How are the payroll taxes of a person working in England and paid by a
US company (located only in the US) determined?

It is my understanding that any payments made in the UK will be
subject to UK Income Tax and National Insurance.

Does US employer WH US taxes?

You shouldn't get taxed twice.

The US has agreements with many countries -- I forget what's it's
called -- but, basically, it's to ensure that folks like you aren't
taxed twice for one income. You have to file in both countries, but
you won't be losing a ton (more than usual) on taxes.

Generally, I think you pay taxes to the country where the income is
earned. -- get an accountant with experience with this because there
are lot of things you can deduct (with the IRS) that you may not be
able to figure out on your own.

Also, check out the IRS website (irs.gov, I think) they have some good info there.

Good luck!

As the employer, I need to know if we need to withhold any foreign taxes or just US.

As an employer, you can obtain guidance from the UK Inland Revenue Help Line. 

(44) 8457 143 143

My guess is that it will depend on whether your employees will be
working here 'on assignment' or more permanently.

You may also wish to enquire whether or not they require a Work Permit
and if they are to be covered under British National Insurance (which
is now part of the Inland Revenue).

If the Help Line cannot answer all your queries, they may be able to
point you in the right directions.