1. As Lizzit says, moving to the UK from the US puts you in the position of the US/UK social security totalisation agreement. This entitles you to remain in FICA for up to 5 years, so no need to pay NI or to get a a number.
2. However once you are working here your earnings become subject to UK tax, so you'll want personal advice on reducing this and getting the 401(k) plan approved in the UK.
Neither PAYE or modified PAYE apply as there is no UK employer. Lizzit is not a UK qualified practitioner so will not be familiar enough with the UK withholding side of things. You do not need a UK payroll company of any kind. You do need a qualified UK tax adviser.
3. The company will also want/need to get involved to ensure that your presence in the UK does not constitute a permanent establishment, so subjecting it to UK corporate tax. In addition the company will need to review its transfer pricing and VAT obligations. Therefore the company will need to do some research and take its own UK accounting advice.
4. Once you are in the UK you will be subject to UK employment law (contracts, maternity leave/paternity leave, vacation entitlement, minimum wage etc etc). All of this compliance adds to the company's burden, so the company will need both a UK lawyer and accountant to organise itself. This cannot be done inexpensively without failures that may cost you or the employer some money.