This phrase always catches me unawares and I imagine the children falling to pieces, but what it really means is that their school term is coming to an end. A school or the people in it can break up, as in:
July 22, the day my kids break up, is ringed in red and underlined in triplicate in my diary: from thereon in, through the long days of August, no piece of work will be safe, no deadline certain, no commission guaranteed. Because if juggling work and children is precarious in term-time, it's 10 times harder in the holidays. (Joanna Moorhead in The Guardian)
The Spring Term begins on Tuesday 3rd January 2006, Half Term is the week of 13th February and we break up for the Spring Holiday on Friday 31st March. (Mayhill Junior School)
How much money have you set aside for your family holiday this summer? Because as every parent knows, once the schools break up, those prices soar, turning a relaxing break into a hefty financial hit. (The Sunday Times)
Another Briticism in these quotations is Half Term (also half-term or half term), a week off in the middle of each school term. There are three terms in the school year: Autumn (not AmE fall!), Spring and Summer, and there's time off at the end of each of these, plus the week in the middle. Working parents spend a lot of energy trying to figure out child care arrangements (or holidays) during these times. Because schools in the US are run by local authorities according to state, rather than federal, guidelines, I don't believe that the names of school breaks are quite as regular. At my school we had various Monday holidays, some time off at Thanksgiving, Christmas and Easter and a week in February that we called the week off in February. These days, I hear more people talking about Fall Break, Winter Break and Spring Break (the last of which is a long-held tradition at universities).
Another thing to note in my friend's quotation above is that she refers to her children as my children, which sounds a little stuffy in American English, where kids has all but taken over informal speech and is not seen as degrading in any way. I have been 'corrected' here by a school teacher who was uncomfortable with me calling her pupils (AmE: students) kids and when I teach language acquisition I sometimes become hyperaware that I'm saying kids, while my students are saying children. As can be seen from the Guardian quotation above, kids is used here, but comfort levels with the term vary.
Kingsley Amis (admittedly a linguistic curmudgeon) complained in The King's English (posthumously published in 1997):
My objection to [kid's] 'committed' use is not to be traced, I hope, to my being snooty, old-fashioned, old or British. No, this use carries a strong hint of being down-to-earth on purpose. It condescends to children and robs them of their dignity in just the same way as it denatures an Italian, say, to call him a wop.
He notes that the 1982 edition of the Concise Oxford Dictionary marked kid as an Americanism, though the US tag doesn't appear in the 1990 edition or any other Oxford books I have. The word has a long history in the UK--as noted in the OED.