The first, from a 1954 letter to Raynor Unwin overviewing reviews of FELLOWSHIP, laments
I must say that I was unfortunate in coming into the hands
of the D. Telegraph, during the absence of Betjeman.
My work is not in his line, but he at any rate is neither
ignorant nor a gutter-boy. Peter Green seems to be both . . .
The second comes a few years later, when Tolkien is pleased that THE ADVENTURES OF TOM BOMBADIL is selling surprisingly well, for a book of verse;
[A&U] have made me an advance, since 'T. B.'
sold nearly 8,000 copies before publication
(caught on the hop they have had to reprint hastily),
and that, even on a minute initial royalty, means
more than is at all usual for anyone but
Betjeman to make on verse!
From this I conclude that Tolkien seems not to have felt any animus against Betjeman and does not envy his success so much as he enjoys sharing in similar good fortune. As for Eliot, Tolkien seems to have largely ignored his existence. Although the two men's work once almost appeared in the same volume,* one gets the sense of contemporaries living in different worlds like, say, Virginia Woolf and Robert Frost.
--current reading: that biography of Fr. Francis (almost done -- thirty pages to go), Tolkien manuscripts.
*Eliot was to contribute an essay on Williams' plays to the memorial volume ESSAYS PRESENTED TO CHARLES WILLIAMS but ultimately didn't have time to do the piece; this is the volume now made famous by the inclusion of JRRT's OFS**
**a piece of Tolkien's that we know Wms liked.