Some of the points made really stood out (italics and comments in brackets mine; bold emphasis and links in the original), although I have omitted a lot of the detailed discussions of traffic risks particular to bicyclists on different classes of roads including special warnings about the fact that railroad crossings in the U.S. are far less safe than in the U.K.
I think you have no sense of how big and how empty the US is. If the US were as densely populated as England, it would have 2.5 billion people. [The true population of the U.S. is about 319 million people and is highly concentrated in about 100 urban areas.] Unless you are in an urban center, you won't find many people of any sort, let alone criminals.
The quickest coast-to-coast route is Jacksonville to San Diego, almost 4000 km, three times the distance from Land's End to John O' Groats, 200 hours of riding for a strong cyclist, six weeks on the road. Here is what it looks like about half-way. See any potential muggers? At about three-quarters. [Do click on the links, the immense granular scope of Google Street View, even in the middle of nowhere, is truly remarkable.] (This is the shortest route, not the scenic route.) . . .
Your big concern will be water. Out west, 100km gaps between sources of fresh water are not at all uncommon, and it would only take a little bit of bad luck (a leaky camelbak shorting out your cell-phone for example) to make dying a very real possibility.
You'll also have to cross two, and possibly three, major mountain ranges. I don't know if there is any route you could take that would not require you to climb to a pass half again as high as the highest peak in England. [This is probably an underestimate for most routes- the highest peak in England is 3,209 feet; I don't think there is any way to cross the Rocky Mountains in Colorado on a road at less than 8,619 feet or in Wyoming at less than 8,323 feet, although you can get a bit lower crossing Montana or New Mexico. Most of the more heavily used passes are higher. I usually cross the mountains at Eisenhower Tunnel which is 11,158 feet.]
Depending on exactly where you are, weather, traffic, vast distances between even tiny towns (often 100+ miles, sometimes much more,) harsh terrain, and/or wildlife will be much larger concerns. Most of those concerns (except traffic) apply more to the Western states. Once you're East of the Mississippi River, things aren't quite as desolate and there will at least be small towns somewhat frequently (though still not nearly as close as they are in Western Europe.)
Make sure to read up on the normal climate of the areas you'll be traversing in the season you'll be traversing them and also make sure you have some way to keep up with the weather forecasts for where you'll be. Weather in many parts of the U.S. (especially the parts away from the coasts) can change very quickly. Depending on season and where you are, the temperature may be anything from +50 C to -40 C. When I was in North Dakota a couple of years ago, it was +40 C while I was there and they had a snow storm about a week later. Even where I live in the Southeast, temperature swings of 20-30 C over the course of a day or two are not at all uncommon, especially during the non-summer seasons. . . .
Also, in case you haven't spent much time in the U.S., it's worth a reminder that the U.S. is very, very large. The distance from New York City to Los Angeles is a little more than the distance from Lisbon to Moscow. You'll be crossing everything from large mountains to desert to plains to hilly countryside. It typically takes about 3-4 days to cross the U.S. by car at 70+ mph / 110+ km/h. Even by airplane, it's similar to crossing the Atlantic. Crossing it by bicycle is not a small undertaking.
A third response:
Apart from a handful of urban areas, USA is far less densely populated than England (and even many urban areas are so sprawling that the urban area of Atlanta has a lower population density than all of England). In remote areas, you'll find roads where traffic is almost nonexistent.
A fourth response:
Wildlife, including dog packs, is more of an issue in the US than in England. This is very situational, in most areas the threat is about zero but if you're off the main roads and in the areas in the west with evergreen forests the bear threat is small but real. Generally they'll leave you alone but getting between mama and her cubs is a very bad idea.
Also, in the southwest the distances can be long and the weather can be hot. I would recommend reading up on desert survival before biking between cities in the summer in the southwest deserts. Unless you're on the interstate you're unlikely to have cell coverage, an accident on a side road could be a very serious situation. Now and then we have cases of tourists who head out into the desert unprepared, have some sort of breakdown and die out there.
On the flip side, winter storms can be a lot worse than you're used to, also. England is surrounded by ocean and the Gulf Stream warms it to some degree. Much of the US is a lot farther from the moderating effects of the ocean and only a small part of it is warmed by ocean currents.
Finally, our extreme weather events are also worse than anything you're used to. If you're near the ocean in the southeast beware of hurricanes. You're looking at winds of at least 120 kph and the worst of them will be twice this. Rain will be extreme, flooding common. The areas near the ocean are usually subject to a mandatory evacuation in such cases and the areas a bit inland aren't but you'll have problems finding supplies and shelter. In the Midwest you'll also have the threat of tornadoes. They are too unpredictable for evacuation, alerts go out when there is threatening weather, warnings when radar sees a tornado and some communities also have sirens. People take shelter where they can, the well prepared have underground rooms. Winds again start at 120 kph and the record holder was 476 kph. All ordinary houses will be flattened by the strongest of these.
Do keep tabs on the weather forecasts, as others have said. It's not just the temperature swings to beware of, there can be downright dangerous weather. Thunderstorms with strong gusty winds, large hail, lightning strikes, and tornadoes are a common thing in the summer months in the middle and eastern parts of the country.
Your two problems are going to be the vastness of the US, an the way our road system works.
The size of it. I had a family friend come from England and stay with us in Florida. They decided they wanted to see the country so they decided to drive to California. They were totally unprepared for the shear distance and what it would mean. Our country is large enough that you have different eco systems as you travel, specially east to west. Your will have Swamp, plantations, flatland, prairie lands, deserts, mountain ranges, arid zones and coastal areas (just to name a few). There are areas that get 170cm of rain a year and others that get 60ml.
This is also true for people. In some areas strangers are welcome and invited guests. In others they are an annoyance to be avoided. It all depends on area. . . .
Even animal concerns are going to be different. Most parts of the US, (even in cities at times) have loads of wildlife. Cougars, snakes, (brown or black) bears, alligator, panthers, crocodiles, bob cats, and so on, are very common in rural areas, some even in cities. Smaller animalas are "worse" if your going to be camping out. Raccoons, Possums, Armadillos, squirels, etc. all will have no problem getting into your food stores.
Cell coverage doesn't reach everywhere are there are large stretches of road with nothing on them. Your could easily find your self sleeping rough.
The English bicycler responding to these and other responses who had originally been concerned primarily about crime states:
I have been comprehensively reassured. You have not only demonstrated that violent criminals aren't a problem, you've demonstrated why as well. What with all the articulateds, trains, hailstones, tornadoes, hurricaines, thunderstorms, blizzards, droughts, earthquakes, bears, wolves, rattlesnakes, dogs, alligators and whatnot, criminals don't hang around because they think it's too damned dangerous.
On the whole the comments made are accurate and sensible. The United States is vast. Huge expanses of the United States are empty farmland, empty mountain forests and deserts. There is far more wildlife, even in cities, than in England and some of it is dangerous. Indeed, even domesticated animals like cows, and herbivores like moose, can pose serious threats. And, we take the ecological diversity for granted.
You forget how inured you become to the ways to safely deal with unmonitored train crossings, extreme temperature changes, hail, torrential thunderstorms, hurricanes, tornados, blizzards, wildfires, floods, earthquakes, sunstroke, and wild animals from mosquitos and ticks to alligators and bears - let alone traffic related risks.
There are crime risks in rural America. Several commentators noted that people can be much more insistent on property rights in the United States and can use guns to make their point. No one pointed out the fairly serious risks associated with hitchhiking these days. One person did point out risks that can be present in the impoverished rural South to someone who looks out of place, and another pointed out the need to be careful in one's political discussions on certain topics in "Red States", and a third pointed out issues particular to Appalachia. Everyone agreed that parts of many U.S. cities can be problematic.
But, since rural America has so few people, and since for a cross-country bicyclist crimes per square mile matter more than crimes per capita, de-emphasis on the importance of violent crime risks in these situations in rural America is appropriate. Predators don't gather where there is very little prey.
The sheer emptiness of much of the United States is one of the important reason why fixed rail transit is more of an economic challenge here than in Western Europe, Israel, India and East Asia.
I recently travelled from Denver to Pueblo, Colorado and back on business (240 miles roundtrip including some minimal driving around downtown Pueblo for a few days). Apart from cities like Pueblo, Colorado Springs, Monument, Castle Rock and exurban to central Denver, most of the trip you pass through absolutely empty desert, even along a major interstate highway like I-25. And, even some of the cities along the way, like Pueblo and Monument and Castle Rock aren't all that big.
It would cost $750 million to build high speed rail from Denver to Pueblo, but it would have to be very fast indeed to make that price worth the increase in speed you could get relative to I-25 where speed limits are mostly 75 miles per hour on flat, straight roads, and a significant share of the traffic is going at 80 to 90 miles per hour. Apart from construction zones and urban rush hour traffic, moreover, the interstate isn't all that crowded on that route. If high speed rail wasn't at least 175 miles per hour, it would be hard to justify, and the cities along the way are quite sprawling, so often you'd need to rely on a rental car, a car share, or dubious and hard to understand local bus service to reach your final destination which limits the benefits conferred by the high speed rail part of the trip.