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, destination, 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.
Note on UK Border Control
In London Heathrow, if you fly from outside the UK or Ireland and leave to another country (neither UK nor Ireland), it is possible to transit without entering the UK and going through UK Border Control, you can stay airside. There is no hotel airside, so if you want to stay in a hotel for an overnight stopover, you have to go through UK Border Control. There is a detailed guide available on Heathrow airport website.
Update 25 July 2015: Visa Seniority
People on any kind of valid UK visa that is senior to a transit visa do not need an additional visa. It means the person can transit on a category "C" (visitor), category "D" (PBS), and so on.
The policy is not available on the net. I wrote to the policy unit using my lawyer hat and got this reply...
They are thinking about including this in the written guidance. Maybe they will, maybe they won't.
protected by Community♦ Apr 22 at 2:40
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