Languages naturally diverge over time, so the "hundreds of years" in your question is actually a cause of British linguistic diversity, not a limitation on it. Every time a generation passes its language onto the next, many small differences build up which cause, like genetic mutation among animals in a population, new dialects and even languages to develop.

It simply doesn't matter much that these different peoples were part of the same political entity: you don't learn your native language from the King but from your parents and peers, most of whom live within 15 miles of you and, especially in pre-industrial times, wouldn't really leave that area more than a few times in their lives.

Adding to the situation is the presence of many English speakers who natively speak another language, such as Scots or Welsh, and learn English later on. Most people speak their second langauges with some sort of accent, and this accent is influenced in consistent ways by one's native tongue.

It might be hard for anyone to answer this question properly, simply because the British situation is normal. You'll see the same thing in other European (and, I assume, African and Asian) countries. Being able to tell which town - or even which part of which town - someone is from is the standard case in place that wasn't "recently" settled. It happens naturally over time.

I actually wrote a paper on this. Specifically relating to Ireland. Ireland is as you may know a very hilly area with some rather rugged terrain for a large portion of the country. This would have made travel very difficult. Meaning that in many cases villages and town had little contact with all but the closest neighbors. Which means that many regional dialects arouse due to such a relatively low amount of travel and outside communication for the avarage person.