As the Roobarb thread says, all of Doctor in the House and Doctor at Large have been regularly broadcast on 7 / 4 Extra since 2004, although not for so long without snippets missing, apparently. If you listen to 4 Extra much, it's hard to avoid them - Doctor at Large is running at the moment! Perhaps a more interesting revelation in the thread is that another Richard Briers series, Brothers-in-Law (Series 1 of which is also currently running on 4 Extra), should have its entire 39-episode run broadcast for the first time on 4 Extra this time round.
The Doctor in the House entry on TVBrain still lists the episodes as missing. Who is responsible for changing it? It seems weird that the program has an entry at all; I for one didn't know radio shows were listed there. Test-searching a sample of a couple of dozen well-known vintage radio comedy series (mostly 4 Extra regulars), I found an entry for just 7 others:
Doctor at Large (complete), Hancock's Half Hour (20 missing), Steptoe and Son (complete), The Likely Lads (complete), Whatever Happened to the Likely Lads (complete), All Gas and Gaiters (24 missing), The Goon Show (lots missing).
There was no entry for The Navy Lark, Dad's Army, The Men from the Ministry, Brothers in Law, Beyond Our Ken and many, many others.
I'll just make some comments on the entries I did find:
Hancock's Half Hour does not include Series 2 Episode 12 'The Matador' (which was recently remade as a 'Missing Hancock').
All Gas and Gaiters includes The Bishop Rides Again and Only Three Can Play, which are broadcast by 4 Extra. Actually all 33 episodes exist, but only 11 have been played by 4 Extra, the others presumably deemed not good enough sound quality. Some episodes are described 'A domestic audio recording exists of the soundtrack, but the programme itself has been lost.' I think it's arguable that if the soundtrack of a radio show exists, the programme is not lost.
As far as I know, The Likely Lads exists only in poor quality (too poor for 4 Extra to broadcast), so, for consistency, if it is 'complete' then so is All Gas and Gaiters.
Where do we draw the line on how good the copy of a programme has to be before we declare it exists?
I don't know whether TVBrain aims to include radio, but just having information that's very partial and inaccurate is going to mislead. TV show information is generally very good in my experience.