I've got to say, I think you'd be taking a major risk just showing up at the airport. Intending to stay for four months in a flat you rent with your boyfriend with kids who will need to go to school, well, it doesn't sound like you're only intending a visit. It may be that you have no intention of breaching immigration rules, but IOs are trained to be suspicious. I'd definitely say you should get prior entry clearance. And, as Weby said, be sure you've got a good insurance policy and private schools lined up for your kids, because using the NHS or state schools as a visitor is a big no-no, and could come back to haunt you if you ever want to settle in the UK. Better safe than sorry.