A progress (or lack of) report.
I have checked the Public Catalogue of the British Library. That shows four WMA publications --- a songbook, a song pamphlet, a book on Irish rebel songs and a book on Soviet songs. I suppose either of the first two might contain 'Pity the Downtrodden Landlord@, but the online index doesn't show titles of items in the books. It would require someone to trot along and look at them in London. Sadly I am some distance from London and don't get there often, so it isn't going to be me.
As a second direction, I located a site for the Alan Bush Music Trust, a charity which continues the work of a former president of WMA. They couldn't help with definitive song details, but they did give me a name and address to write to.
I shall now keep you all in suspense by not naming the person I have written to.