If you work in Soho/West End you have a variety of neighbourhoods to choose from depending on your budget and what type of area appeals to you. North, South, West, East.
Islington, Crouch End, Muswell Hill, Camden, Belsize Park, Hampstead, Highgate, Clapham, Dulwich, Battersea, Chelsea- Kensington, Notting Hill, Wandsworth, Balham, Earlsfield, Tooting
All good for families and have green space for dogs. Also they are accessible to the West End. If you need to get to the school for any reason from work it will be much easier than if you live further out.
I've done what you are doing the other way (to NYC) and then came back again. Since you will have missed the State School application deadline and you have 1 child reception age, you may need to consider private while you are here. Good schools here are notoriously over subscribed (like NYC) and you will be at the very end of the queue once you have an address.
Equally it can be tough to find a reception place in a private school. People put their kids names down here at birth for some of them.
I'd find a school that will take them 1st, then find your accommodation. You want the kids to be able to get to school easily. Walking if possible or a short bus/bike ride. Start phoning various schools, but check their Ofsted ratings and reports to get a feel for them first.
I'd also look at the international schools. They are usually more willing to take on situations like this but it is £££ expensive.
If you can get them into an English School I'd do that first. It's a great experience for them.