Can ESTA holder do some remote work from US for a UK company? [duplicate]

Even if it is not legal, enforcing such is downright impossible. You sit at the laptop and type. The money is paid out in the UK to a UK account. There's exceptionally little the US authorities can hold on to.

No, it's technically illegal, any sort of work in the United States requires a visa. From the B-1 criteria defining "business", which apply to VWP holders as well:

This covers a wide range of activities such as attending meetings, consulting with associates, engaging in negotiations, taking orders for goods produced and located outside the United States, attending conferences, and researching options for opening a business in the United States (such as locating or entering into a lease for office space). Generally speaking, you cannot engage in any activity or perform a service that would constitute local employment for hire within the United States.

That said, as @chx points out, this is virtually impossible to enforce. The main catch is going to be on arrival: you'll be asked why you're coming to the US and what you intend to do, and answering "working locally" is not going to fly. "See your girlfriend" is also a somewhat risky answer, since they'll presume you're trying to move to the US permanently.

So I'd advise you to come up with a permissible reason. "Training", for example, is generally allowed, and isn't even a lie if you're doing "on the job training".