They are allowed to live, work, and study in the UK without any significant restriction, and the tax issue would be a non-issue if they were domiciled in the UK. The reason it's interesting and complicated is because they're apparently living and working in Germany as an employee, without FogCreek having a German entity. They wouldn't be subject to British taxes and there's no German business entity to register with the German tax office.

A UK company (or a company of any jurisdiction) can establish a subsidiary in another EU state, in much the same way you could register a NY-incorporated company as a foreign company in CA, but this doesn't appear to be what's happened in this case.