Is there any way to determine whether or not a film has been broadcast on a TV station (particularly in the UK)?
For the UK:
The BUFVC has an archive of UK TV listings back to 2001. You need to be a member to access the full archive, but you can search the previous 2 weeks for free.
Alternatively, you can try The Guardian's TV Guide it's a bit hit and miss and it only tells you what channel the film was last broadcast on. For example it will tell me that The Shawshank Redemption was on ITV2, 12 Angry Men was on Sky movies crime & horror and that Rollerball was on TCM2.