The Gambling Commission publishes statistics on the number of bookmakers in Britain. Earlier this year they recorded 9,021 bookmaker premises, based on returns from operators. The number has been fairly constant in recent years.
A data user will be able to find about 1,280 of these in OSM (depending on how determined they are).
The preferred tag is "shop=bookmaker". That picks up 8% of all bookmakers in Britain. The most common alternative is "shop=betting" and that will pick up about 4% of the total.
There are a number of variants on these (shop=bet, bookmakers, betting_shop, bookies, turf_accountant) Together these pick up about 0.4% of the total.
A few contributors have gone down slightly different routes. Using the "amenity" tag rather than the "shop" tag accounts for another 0.2%, and various bookmaker-related values for "gambling" account for another 0.4%.
There are also some premises which look like a bookmaker (based on the operator or the name), but aren't marked as such. I found another 79 premises that might have added to the collection if they had been tagged with additional data. There are almost certainly more that I missed, but deep searches to find these get increasingly complicated and the results increasingly suspect.
I'm unable to find around 86% of British bookmakers in the OSM data.