That's a difficult question to answer. Who should we try to count? Everyone who can ring plain hunting on a coursing pair? Everyone who can ring Plain Bob Minor from the trebles or tenors? Everyone who rings handbells regularly?
Instead we can answer an easier question: how many people have rung a handbell quarter or peal, published on BellBoard, during a certain period?
First I looked at the year from 1st July 2012 to 30th June 2013 (with the help of a computer program to collect and analyse the data). I found that during that period, 292 people rang at least one handbell peal, 485 people rang at least one handbell quarter, and 572 people rang at least one handbell peal or quarter. This means that 87 people rang only peals, 280 people rang only quarters, and 205 people rang at least one of each.
A convenient source for statistics about peals (but not quarters) is PealBase. Looking back over the last few years, the number of people ringing handbell peals each year is around 300, and the number of people ringing tower bell peals each year is around 3000. The total number of peal ringers per year is also around 3000; I think it's safe to say that almost everyone who rings handbell peals also rings tower bell peals. These numbers are consistent with my earlier observation that the number of handbell peals rung is around 10% of the total number of peals rung.
A common estimate of the total number of ringers is around 40,000. I don't have a source for this estimate. In that case it seems that the number of peal ringers each year is around 7.5% of the total number of ringers. Assuming that the same proportion applies to handbell ringers, we would get an estimate of 4000 for the total number of handbell ringers. I think this would be a huge over-estimate: we know that there are many tower bell ringers who only ring at their own tower, rarely ring quarters and never ring peals, but handbell ringing is organised very differently.
Going back to the data from BellBoard, we have a first figure of 572 for the number of active handbell ringers, defined as people who rang at least one quarter or peal, published on BellBoard, during the year ending 30th June 2013. But maybe a year isn't a long enough period to get an accurate picture. As I looked down the list of 572 people, there were several notable omissions just from the relatively small group of ringers who I know well: people who are good handbell ringers but happen not to have rung anything publishable during the last year.
(I should say at this point that my numbers for quarter peal ringers are not completely accurate. Although peal ringers generally use consistent names for publication, there is a remarkable amount of variation in the names appearing in quarter peal reports, and it takes a lot of work to clean up the data; eventually I accepted that I would not be able to correct the names of absolutely everyone. Even looking at the variation in peal ringers' reported names, I can see that Andrew Craddock does an amazing job of keeping PealBase consistent.)
Anyway, back to the data. The next thing I did was to look at the 5 year period from 1st July 2008 to 30th June 2013. I didn't separate out the quarter peal ringers and peal ringers, but I found that a total of 1059 people rang at least one quarter or peal. That's almost double the number from the last year, which I found surprising. So then I looked at the 10 year period from 1st July 2003 to 30th June 2013, and found that a total of 1306 people rang at least one quarter or peal. I decided to stop there, because online quarter peal records aren't available for much further back.
My conclusion is that there are around 1300 people who have rung a published quarter or peal on handbells in the last 10 years. I won't claim that this is the right figure for the total number of handbell ringers. For example, it doesn't include any of the children from the Mount Vernon Handbell Club, or the hidden handbell ringers that we discover when we take bells to SACR meetings, or people we are teaching who are not yet up to quarter peal standard. But I think it's about the best figure we can get from published data.