Every poll I have seen shows support above 70%. The highest I've seen was in the recent golden jubilee year around the time that the Queen mother died, that was 82%. It goes up and down, but tends to stay above 70% and support is different for every member of the RF, with the Queen obviously having the highest support.
The break down in Britain:
Loyalists in Northern Ireland will obviously be strong supporters. In certain areas of Norther Ireland you can actually see where people have painted giant murals of the Queen on the side of buildings, however NI also has a large republican community that is also very vocal (sometimes violently so).
Northern Wales is more loyal than southern Wales. You'll find a lot of republicans in the south, but the farther North you get their numbers will dwindle.
Scotland tends to have a lot of republicanism, but I think it's probably stronger in the cities and then eases up in the country.
England has its pockets of republicanism, particularly in the North and in the cities, but the country side tends to run loyal.
I'd say two thirds of republicans are simply stupid and just don't understand how the government works or what the head of state does, you can find some examples of this in these answers. Many people don't seem to understand that if the monarchy were abolished we would still need a head of state, but that they would be a politician who would still need money to pay for head of state expenditures. The money the Queen gets wouldn't all of sudden go to hospitals if she quit, it would just now go to a politician. The other non-ignorant republicans are those that want to be able to vote for their head of state, which while I disagree with is at least valid reasoning.