What is the equivalent British English word for American homebody?

There isn't a direct equivalent to "homebody" in British English. However, you could use these phrases depending on the context:

* Stay-at-home person: This is a neutral and common way to describe someone who prefers to stay home.

* Home bird: This is a more informal and slightly whimsical term.

* Housebound: This emphasizes the act of staying home, especially due to some limitation or preference.

* Domesticated: This implies a preference for the comfort and familiarity of home.

The best choice will depend on the specific situation and the tone you wish to convey.