I currently work a 9-5 job in the UK but as I work remotely could do this job from anywhere, so my question is - am I allowed to travel to the US under the VWP and continue to do my current job, on my current salary, continuing to be paid into a UK bank account, with no intention to stay more than the maximum 90 days than the VWP permits? Or would I still need a work visa to do this? Thanks in advance
Not sure this really qualifies as an answer rather than just a comment, but here goes...
This question has been asked quite a few times for various countries, and it is generally considered a gray area which was not anticipated by lawmakers when immigration statutes were drawn up (as MadHatter points out, Canada is a notable exception).
In general, you are not allowed to work on a tourist visa. In addition, they could be entitled to tax you on that income (though in theory only if you are resident in the US, i.e. stay there over 6 months in a year, but the rules are complex).
Working a bit while you are on holiday or doing business is one thing. Working full time for 3 months while you are in the US is probably over the line.