Hi Marie,
As simple as it sounds try Typing 'Bournemouth Wedding Bands' into Google and the first results you should get are from Google Maps. I've looked and these seem to have quite a few listings.
I wouldn't limit yourself Geographically as you may find some great bands at a reasonable cost that are willing to travel.
I know that it's tricky looking for wedding entertainment but it's just about knowing how to search and where.
If you're really struggling and simply don't have the time to search in this way you may want to try some entertainment agencies. Here are some popular ones that I have dealt with or known of in the past. You can be quite specific with them about what you want but beware! as they do charge a commission fee. Get them to disclose how much is going to the band and how much to them:
I hope this helps and that you find what you're looking for