Ironically, he's not singing in F initially, nor F♯ when he sings that he is.
The song's far more in Dm, and the F♯ note he sings is actually the major 3rd of D - yes, it's an F♯,, but he's certainly not 'singing in F♯. Which limits the funniness of the piece, unfortunately.
EDIT: trying to find why musos may find this funny. There have been many occasions when I've encountered singers who've said 'I only sing in F♯ (or whatever key)'. Most musos will understand that it's actually a statement which doesn't hold water. For a couple of reasons.
Songs' range can vary depending far more on their highest and lowest notes, which are in fact independent to the key they're in.
Most folk have a range of more than an octave, whilst a lot of songs span only one. So changing key, up or down, will be possible for most singers - albeit a tone or two, so 'only in F♯' doesn't ring true.
Les Dawson (British comedian) was an accomplished pianist, but he usually incorporated in his act renditions of popular songs, which he expected the audience to join in with, but spectacularly made horrible mistakes with some notes - very cleverly worked out - while retaining his inimitable smile, as if there was nothing wrong. Now that was funny!
Tim has (probably) taken this and the first reason to write (?) and sing the song. But from maybe a slightly different viewpoint. He sings those F♯ notes against the F♮ played in the D minor chord, which is discordant. Had it been the other way round, it'd just sound Bluesy. Hence the reference to F♯. Musos would appreciate this, and lots would think it was actually difficult to do - it's not, but it's effective. But I still say it's a misnomer!
EXTRA EDIT: it'll probably resonate with some musos my experiences when a singer says 'I sing in x key'. You play a chord for that key, and they then proceed singing in a completely different one. Yes, sometimes F♯...