Interesting - it does suggest that the Red Flag was fluttering in Tolkien's subconscious, though I can't imagine him singing it out loud with any enthusiasm. His politics, insofar as he revealed any, were a kind of gentle anarchism - there's a wise saying, I think by Gandalf, on the lines that the world can't be made better by 'great men' or their projects and programmes, it's only millions of quiet, simple acts of kindness by ordinary people that make any real difference.Reminds me of:
Far over the misty mountains cold
To dungeons deep and caverns old
We must away, ere break of day
To seek our pale enchanted gold
(From “The Hobbit”)
Edward Thomas's 'Yes, I remember Adelstrop' fits the same tune. And a great many other ditties do too.