Most big movers (Allied, Mayflower, United Van Lines) do international moves. Call at least 3 different companies to compare quotes
The amount depends how much you are shipping and how quickly you want it. Shipping only boxes by sea and getting it in 6 weeks a lot cheaper than shipping all the contents of a 3 bedroom house and you want it
quickly. It also depends if you want to pack yourself or have them pack, if you need to buy boxes and packing material, if you buy insurance and the value of your items, etc
But all movers will give you an estimate. Stick with the big companies who are used to international moves and have all forms
I used Allied to move to Europe but that was 7 years ago. I only had a tiny one bedroom apartment and my furniture was a bed, sofa and armoire. I left behind the cheap Ikea table and chairs, bookcases, etc as it would cost more to ship them than to just buy new/used furniture once I got settled. I used Allied because that's who my company had a contract with. It was a big company who transferred staff nationally and internationally and they were happy with the company. I had no problems. I don't know the cost as it was billed directly to HR as they paid for my move
There is no Ancestral visa. There is an Ancestry visa, but that is for citizens of Commonwealth countries only.
Wanting to move and actually being able to move to England are two completely different things. Do you have the necessary visa? If not, get that before you start worrying about moving companies.
We have what's called an Ancestral Visa