Happy holidays is an American seasonal greeting--one that I find very useful, because (1) I don't like to say (AmE-preferred) Merry Christmas or (BrE) Happy Christmas to people unless I know for sure that they celebrate Christmas, and (2) it saves having to say Happy New Year as well. To Christmas-celebrating Americans, 'the holidays' are Christmas and New Year, but the phrase happy holidays could equally refer to Hanukkah and New Year or Kwanzaa and New Year or Solstice and New Year, etc. So, it's an equal-opportunity phrase.
But over on the UK-Scrabble list, I've just discovered that some people (or at least a Dutch person writing in) believe that the holidays in Happy Holidays refers to Christmas and Boxing Day. Not the case. Americans do not observe Boxing Day.
Better Half and I can think of no BrE phrase that people actually say that does the same job. In cards, one sees Greetings of the festive season or Season's Greetings. In speech, people just say Happy/Merry Christmas. Boxing Day doesn't get a mention. A few people do say Happy Holidays, but this is perceived as American and often resented for being "politically correct". To me, politically correct just means 'polite and considerate', so it's beyond me why it should be resented.