If your children are in the US curriculum they will likely struggle crossing over to the UK curriculum. I move US families professionally to the UK, and find most children settle / transition best into the American schools where the curriculum is consistent or flip to IB. Below are those I find most popular, catering to a wide range of needs.
www.acs-schools.com/ - they have several Campuses.
www.tasisengland.org/
www.asl.org/
www.southbank.org/
Marymount is another popular choice, depending on the age of your children.
Most have excellent bus networks, but you'll want to think about their commute time, and triangulation of home / office / school as travel across London can be very time-consuming.
If you opt of a UK curriculum school, you'll find most schools will be more than willing to advise you on tests. Some use the Common Entrance, others have their own tests. We did Common Entrance using Galore park resources for study and my daughter did some past papers. The past papers are a little pricey but eminently worthwhile. ISEB is a great resource.
In respect of taxes, it sounds as if you're being moved as an expatriate by your employer vs a 'local' basis. If an expatriate, and school fees are funded by your employer, this benefit would commonly be grossed up for taxes i.e. your employer takes the hit on the US taxes due, as well as any UK taxes on benefits / sourced income / tax gross ups etc. It would be very unusual for the employer to pass any tax burden to the employee under a true expatriate arrangement. Your liability ought to be limited to US federal & state taxes / FICA etc on employment income ( i.e salary / bonus). If on the other hand you're moving on a local basis, you will need to consider that at least for the first year you may have continued residence in the US, in addition to assuming UK residency. Beware that dual taxation can be an ongoing pain point for US citizens and / or green card holders. I specialize in expatriate arrangements, so feel free to PM me if you have any questions!