From a UK perspective the word "contractor" and the indicia of self-employment have a very different meaning from within the United States. If you are likely to be treated an employee you would be entitled to a UK employment contract, leave allowance, maternity/paternity pay, sick pay etc under English (or Scottish) law.
Your existence might constitute a permanent establishment of the US company so requiring filing of UK corporate tax returns.
The turnover may be subject to VAT.
You will still need to pay UK tax & NIC on the income.
Generally speaking it is best that the engager of the services takes advice in the UK so that it is regulated in the UK, which is where the work will be carried out. Advice of the kind described would always be best paid for by the engager of the services as they are taking much of the risk.