Usually. Some very talented voice actors can do it. But generally they just use Dick Van Dyke Not Quite Cockney or an exaggerated RP.
I wouldn't like to say how many accents there are - it depends on the hearer. I can tell the difference between Barnsley and Wakefield, but a southerner wouldn't be able to tell the difference between any Yorkshire or Lancashire accents.
Main groups are probably:
Scots
Welsh
Irish
North East ("Geordie" - IME Cumbrian accents and some NE Yorkshire accents are of this type)
Northern - Lancs, Yorks, East Midlands
Liverpool (Scouse - distinct from other NW accents)
West Midland - (Brummie)
East Anglian
South Western (Get off moi laaaaand!!!)
South East (incl. Cockney)
RP - not regional. Spoken mostly by ex-grammar school people and public/private school educated individuals.
Depending where you come from, you will see different subgroups. An Essex girl sounds pretty similar to a Kentish Maid to me, but I suspect someone from round there would say "rubbish - they sound completely different".
For the record, there are only three US accents I can distinguish. The one used by Homer Simpson, the one used by Cletus the Slack Jawed Yokel (isn't that a southern accent?) and the one used by Bill Gates, makers of IT training videos, and Kermit the Frog.