It's a cross between baseball and croquet that british people can play while they sip their tea

It doesn't much resemble croquet, except in the very general way that yes, they're both entertainingly daffy ways to spend a long English summer's afternoon.

Do brits play basketball, football, or hockey?

Basketball: not really, but a similar game (netball) is popular in schools and colleges. It doesn't get much media attention.

Football: means "soccer" to a Brit. American Football does exist, but isn't a big media sport. Some friends of mine at university were in the uni team, but this was Cambridge and they'll play any old sport there as long as there's a chance they can beat Oxford at it. Average crowd turnout for the matches (on Coldham's Common, not in any kind of stadium) was well under 10, although the Varsity match (Oxford vs Cambridge) drew a crowd of about 30. As for the US American football leagues, the Superbowl result will get mentioned in media sports bulletins, but that's about it.

Hockey: ice hockey no; "hockey" to a Brit means field hockey, which again is popular in schools and colleges but gets no media coverage. Despite occasional forays into more exotic sports, school P.E. lessons still centre in the summer on cricket for boys and netball for girls, and in winter on football (soccer) for boys and (field) hockey for girls. And sometimes rugby for the boys if they're extremely unlucky.

The only really huge media team sports in Britain are football (soccer), and, to a lesser extent, cricket and rugby. AFAIK there are no professional basketball, netball, American football, ice hockey, or field hockey teams in Britain.

Peter