London speech is not dissimilar to the accent used by by millions of native speakers throughout the whole of the south of England. It's also an accent that we hear all the time on TV, radio and films. People from, say, Manchester or Glasgow speak very differently from Londoners, but we are all so used to hearing southern English voices in the media that this accent presents very little problem for any of us. ( The same goes for American accents, by the way - constant media exposure means that we all understand American English without any trouble, but that's another story).
As Kate says, it might take a moment or two to attune our ears to the speech of another region of the British Isles : for example, if we switch on to a TV drama set in Liverpool or Dublin, we'd have to listen harder for a few moments. We might even miss a word or two if the speech is rapid or the language is colloquial, but we would still understand what was being said. And if there is an odd word which we don't catch, we can always infer the meaning from the context.
English is not like Arabic, which has a totally different and sometimes mutually incomprehensible dialects in different regions. Nor is the UK like Italy or the German-speaking countries, which have dialects with different grammar and vocabulary. The English spoken by most people in the UK is actually the 99% the same wherever you are - it is only the accent which varies. We have all heard these accents before, so it's simply a case of tuning in the right accent.