If you're flying TATL Business class, I doubt it makes a difference; I've not found any TATL business flight on which anyone cared how many, or what size(s), my carry-on luggage was. I've flown Delta, KLM, British, American, United, Virgin and SAS. None of them showed the slightest interest in my carry-ons. The overhead compartments are usually huge in business class.
Euro-business (internal Europe) is a different matter entirely, and if you have flights that both originate and terminate in Europe (including the UK), the rules usually need to be followed pretty religiously IME.