Is there a website or some other way to find out if I need a transit visa for a short stop in the UK on my way somewhere else? Ideally this should give me the answer for all nationalities, and take into account any other visas I may have.
Why yes there is.
This UK government site will tell you if you need a visa to transit through the UK. You enter your nationality, desination, and any visas you already hold, and it will tell you if you need a transit visa or not. It also includes instructions for applying for one.
As a summary (but check for exceptions):
If you need more information than is provided by this website you will need to check with your airline or contact the UKVI. The Home Office's Charging Procedures - A Guide for Carriers also provides some guidance on the subtler points of the rules (it's intended for professionals, not passengers, but does explain in great details when a visa is or is not required).