Why is your company paying such a ridiculously low salary? That's a major red flag for this whole thing. That's not just kind of low, but insultingly low and bordering on not even meeting the living wage. It's also an indicator that either they don't care about you at all, or haven't planned this position at all and will be equally incompetent about every other aspect. This must be addressed before anything else. I would go to one of the many job finding websites (jobserve if it's tech) and get a feel for what similar jobs to yours will be paying IN LONDON. Undoubtedly it will be significantly more and you can use that as evidence for your demand to double that salary. Think about that, even with the salary doubled you will be in the range of typical London salaries, probably on the low end if you have enough skills for them to move you to London.
I guess you could do it, but why bother? You will either have to live in a place outside of London that's smaller and crappier by far for what you could get in the US and commute for 1.5 hours each way, or live in the cheapest, most crime ridden terrifying part of London. You won't have any money to travel, take holidays, eat out or even go to the pub. Or even buy clothes for your kids. You won't be able to afford a car, so if you live in the country you won't be able to go anywhere. Sure you can use public transport in a city and make do without a car, but it's a different matter in the country.
The only way any of this makes sense is if you can work from home almost all the time. As that's becoming the new normal, maybe that's what they are thinking. In that case, everything is completely different as you can live lots of places that are much, much, cheaper than London and only come to the office once or twice a month. That's all you will be able to afford anyway as Margo's £35 trip to London is no exaggeration. In this case, I'd think about somewhere like Margate, or maybe Lewes where you could actually find something for £1000 per month.
One more thing, childcare will take up ALL of your wife's income if she chooses to work. I hope she's ready to be a stay at home mother with no money , which will be pretty bleak when it rains all winter and gets dark at 4:30. And even bleaker when you get home at 7:30, exhausted from a hell commute, ready to wake up at 5:30 to do it again.
Sorry I couldn't find any humour for you!