Hi Mike,

I have mp3s of Carroll Gibbons 'All Or Nothing At All' and 'How Sweet You Are'. I'll email them across when I get the chance.

A couple of the 1943 tunes are on the internet archive:
Carroll Gibbons - Pedro The Fisherman (here)
Eric Winstone - Tenement Symphony (here)

Two others are currently for sale on ebay:
Joe Loss - Number Something Faraway Lane (here)
Turner Layton - If I Had My Way
(here)

Eric Winstone - 'I Never Mention Your Name' is apparently included on this box set (see discogs release here). There's currently one for sale on ebay (here)

And finally, there has been a large influx of new 78s uploaded to the internet archive recently (and continue to be uploaded). A lot seem to be Australian releases, and among them are two of the accordion band tunes you've been looking for:
London Piano Accordeon Band - Hey Little Hen (here)
Eric Winstone - Ma! I Miss Your Apple Pie (here)
It's probably worth keeping an eye out as they continue to add more over the coming weeks.

All the best,
Peter