There are an awful lot of assumptions on this thread.
Firstly, there is huge variation in private school fees. Yes, some may cost in the region of £35k-£40k per year, but there are plenty more in the £12-15k range, especially in some parts of the country.
Then there is the possibility that the boyfriend's dc may receive a significant bursary. Again, the availability and size of these varies enormously, but some schools would certainly award bursaries to parents on that kind of income.
Thirdly, not everyone lives in London or the South East. While it would be tight living on what's left over after paying school fees, it could be perfectly possible in some parts of the country. We also don't know what the boyfriend's living situation is, whether he has a mortgage, or how much he has in savings etc. He might have been saving up for this for years for all we know.
There is the assumption that he must be naive/crap with money for even suggesting this, when he might have a perfectly viable plan.
And finally, there is the assumption that the OP is on a low income just because she is part time, that she must be relying on state benefits and that she is looking for a man to fund her lifestyle. Again, none of those are necessarily the case.
Do people not realise that people's financial situations - income, outgoings and savings - might be significantly different from their own?