Unfortunately, those who may know the real reason as to why this might be important haven't replied. So, I'll play the straw man. I had a the same situation years ago.
You signed a contract when you began work. Is the company you signed with registered in Companies House as a UK company? Does that UK company control the UK payroll, UK pensions, and UK tax? Does it stipulate in your contract it is in accordance with English Law? Mine did, and I concluded it's a UK company and foreign for US tax purposes. When stock options came about, the US side insisted I (incorrectly) sign a W8-BEN, so they saw us as a foreign employees. It is not "both" IMHO. Office procedures being performed in a "US way" mean nothing for tax purposes, IMHO.