So do I need to have access to a vehicle before starting lessons or can I be taught in a company car?
Legally you can take lessons in
any car, so long as it is roadworthy, taxed, you are insured to drive it, etc. Instructors normally have their own vehicles for lessons, but some will take you out in your own car if you wish. I'm assuming that's what you mean by a "company car," and not just a vehicle owned/supplied for you to use by the company for which you work?
Can a person without a license buy a car?
Anybody can buy a car, even if not old enough to be eligible for a license. Anybody, of any age may also
drive a car without a license, so long as it is on private land to which the general public has no access.
If I hold a US license but am a UK national, do I still have to surrender the US passport?
I'm not sure, but presumably you have a U.K. passport as well which could be used for ID?
You will take your lessons and usually your test as well in your driving instructor's car (I've never heard of anyone not doing this
Lots of people learn and take their test in a private (non-instructor) car. At least they always used to -- Maybe it's become less common?
But there are no legal restrictions either way.