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.
HOWEVER: There have been changes taking effect Dec 1 2014 to which countries' nationals require transit visas. Regardless of the link above, check this page for a revised list of which nationals need transit visas if they are passing through immigration (VIT - section 2) and which require visas if they pass through immigration or remain airside (DATV - section 1). Procedures remain the same.
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).
Changes to the rules activating on 1 December 2014
A Statutory Instrument entitled "The Immigration (Passenger Transit Visa) Order 2014" will make large portions of the above text obsolete. Please refer to the instrument itself at http://www.legislation.gov.uk/uksi/2014/2702/made until further guidance becomes available. The government's interactive page, which leads the user through a series of questions will take some time to sync up with the legislation (if ever) and should NOT be relied upon after the 2014 order takes effect.