"Is that a realistic salary for a Java developer?"
Yes. That sounds ballpark for mainstream Java dev work with that kind of background.
"Also, if it is, are there any chances of earning more?"
Work for a bank. Seriously. You could maybe get to 110k-120k. I've interviewed for 200k roles with hedge funds/similar, but I don't have the maths to pass them. Or in one case the solid decade of D3 they decided they needed after dragging me all the way down there...
"I'm also wondering how much value the £4K-£4.5k can have in London, for my family with 2 children?"
You're about to be upset... A 3bed flat in a reasonable area runs anything from 2-4k a month. Food, beer, transport -- all expensive. You're unlikely to find much to buy with a 250k mortgage ceiling. You may find you can live better in Oxford for a lower salary (don't forget that you're paying 40% + NI on any notional bump and when you get even higher the tax rate grows[1]). Also consider QoL for kids in London.
The upside is that the London market is what people would call "buoyant"; which basically means you'll be sat next to empty desks that can't be filled, you'll interview candidates a lot and can walk out of the door straight into another place if anything goes wrong because everyone is hiring all the time. Once you're there, it's all a bit heady.
[1] I have payslips with 59.7% deductions on them and that's pretty depressing when you're jammed into a tube at 07:00 in the freezing cold dark winter listening to someone's crappy headphones going "Tsssst, tsssst, tsssst, tsssst, tsssst, tsssst, tsssst, tsssst, tsssst, tsssst, tsssst, tsssst..." for an hour while TfL intermittently shouts "kkkkkrrrkkkk-orry kkkkkrrrrk-elay krrrrkkkkkk-ignal failure at kkkkkkrrrrrsh" over a tannoy last serviced when Napoleon was breathing.