This is essentially a question of perception and politics. Simply counting 'named' accents or doing mutual comprehension studies will get you certain numbers but will not answer the real question: What do accents, dialects, varieties and related 'minority' languages mean in the area you're studying.
You will find some broad similarities such as the creation of prestige levels, standards and historical change. But you will also find a lot of differences in how the prestige is managed (e.g. what do you hear on TV, how pejorative names are used to label accents, how accents are recognized as being part of an identity vs. poor learning, how standards are established, etc.)
I would suspect that the difference between English (in England) and other major European languages is that the current English standard is relatively weak and the the formerly high-prestige accent is relatively low prestige now. You also have a lot of political resurgence of local identities which has led to a relative increase in accent prestige (though not uniformly and not all). You also have relatively high awareness of accents which is broadly reflected in popular literature (My Fair Lady, etc.)
My fleeting impression is that in French and German, the standard is relatively strong but is not necessarily as strongly associated with class. Both French and German have many very diverse 'dialects' some of which may have their own literary forms. Contrast that to Russian where accents are not very strongly linked to local identities - perhaps with the exception of Moscow. The language where I can make observations on the same level as English is Czech which has a very strong standard which is much more strongly differentiated from non-standard based on morphology than accent. The only two widely recognized accents are Moravian (with a strong subset of Ostrava region and Brno) which is strongly linked to identity and Prague which is very weakly linked to identity. While there are other regional varieties, they are much weaker both in recognition and identity. The levels of mutual intelligibility vary from almost full to relatively low.
So looking from the perspective of Czech (1/5 of the speakers), UK English will seem much more diverse and the networks of decisions speaker and listeners have to make when assessing someone's identity based on their speech much more complex. This whether or not the actual number of varieties per area or population would vary. I would estimate that coming from Russian, you'd get a similar perspective but would not dare to guess at what English would look like coming from French, German, Spanish or Italian.