I've had a look round and whilst some councils list bridges on their websites, I can't find a uk wide one. You can get The AA Truckers Atlas which shows all restricted heights on A and B roads. If you want to know for a particular one off route, put the details on here and I'll check my map for you.
thanks people very much for your answers, and sddsdean that is most kind of you and I appreciate your offer. While I do have the Truckers Atlas it seems to be the only publication detailing road bridges heights . I may well take you up on that offer !