As far as I know: your US prescription will not be accepted in UK pharmacies since American physicians are not licensed to practice medicine in the UK (and vice versa). Assuming that you are taking up residence for 3/6 months minimum you can register with a GP as a patient on the NHS, that is the National Health Service, once you have a 'permanent' address and request a referral to a rheumatologist which could take 2 or 3 months to come through. Entitlement depends on residence not nationality. I rather doubt that a GP would write a prescription on the say so of an American doctor.It would be a good idea to organise a 3 month supply.
In case of emergency you will be treated on the NHS whether registered or not, although a fee might well be required
You can see a rheumatologist privately usually within a couple of weeks. Your GP will normally be willing to prescribe the medicines the private specialist recommends which you can obtain on the NHS with a 'copay' for each item prescribed. I think that's about £7 these days but not sure.
If you get a private prescription filled from a specialist you have seen privately, you have to pay the full cost of the medication.
People coming to work in the UK for US based employers who have a US health scheme through their US employment, usually have access to private insurance in the UK, either with membership of a UK private health insurance company such as BUPA or reimbursement of private health care costs. by arrangement with the US Health inusrance provider.
It is in the employer's interests to have their personnel as fit as possible to work thus getting medical care as soon as possible and avoiding the long delays that are common on the NHS. Many large UK employers offer private health insurance as part of the benefits.
The best known UK NHS lupus treatment centre of world wide renown is St Thomas' London although there are other centres of excellence such as King's College London. The London Lupus Centre is a new private lupus treatment centre staffed by the former chief of St Thomas' Lupus Unit, Dr G Hughes and many of the top specialists who also work at St Thomas' and others. See their website.
Either way, you should have a letter from your current rheumatologist giving details of your case and current medications.