Your law firm should have you on an expat package, which should include:
tax equalization
tax preparation for US and UK
clever tax breaks for Resident but Not Ordinarily Resident status in the UK.
If your company is doing a tax equalization programme, then it does not matter what you do to reduce your UK tax, you will never pay more than your taxes would have been had you remained in the US. In this case, to reduce your tax you should do the same things you do to reduce US tax: keep track of your US charitable donations, US income and sales tax paid, mortgage interest paid, school loan interest paid, business expenses, maximize your 401k contributions, and so forth.
If your company is not doing a tax equalization programme, then it makes sense to save the maximum amount of UK tax dollars. If you are on the US payroll, contribute the maximum to your 401k programme. Open offshore bank accounts for your direct deposit, which may help to allow you to shelter your offshore workdays from UK tax. It also shelters the interest earned on the account from UK tax. Be sure that UK PAYE (income tax withholding) is being done rather than US. Be sure to stay on US SS if you can; the US system is generally understood to be better. Your housing costs may be deductible, depending on a number of circumstances, so keep track of those as well.
If your company does not understand any of these issues I just raised, they should seek expert advice to sort you out.
Always be prepared to pay the most tax; that way, you are pleasantly surprised when you pay less. It is more stressful to go the other direction.