What is the British English equivalent for "homebody"?

For what it's worth, the Longman and Collins dictionaries, both British publications, do not mark it as Americanism (although Macmillan does). The OED marks it as "originally U.S." but finds examples of it in The News Chronicle (London) from 1959 and even in P.G. Wodehouse:

He said this girl was one of those domestic girls, a little home-body, and might be leaving the party any moment now. — Bill the Conqueror, 1924.

For what it's worth, Roget's Thesaurus does not have an entry for homebody, and the synonyms suggested in the Merriam-Webster thesaurus are all much stronger and mostly negative compared to homebody. Someone who prefers to stay and entertain at home rather than going out to bars or events is hardly withdrawing from human society or avoiding human contact in favor of a life of solitude, as implied in terms like recluse, loner, hermit, shut-in, or anchorite. They might well be insulted by the characterization.

Just say homebody if you mean homebody.