Can a US citizen live and work remotely in the UK for a month?

A key point in this situation is that UK Immigration Rules on working while in the UK as a visitor state that visitors must not receive payment from a UK source for any activities undertaken in the UK (V 4.7 https://www.gov.uk/guidance/immigration-rules/immigration-rules-appendix-v-visitor-rules)

The other relevant rules are:

V 4.5 The applicant must not intend to work in the UK, which includes the following:

(a) taking employment in the UK;

(b) doing work for an organisation or business in the UK;

(c) establishing or running a business as a self-employed person;

(d) doing a work placement or internship;

(e) direct selling to the public;

(f) providing goods and services;

unless expressly allowed by the permitted activities in Appendices 3, 4 or 5.

V 4.6 Permitted activities must not amount to the applicant taking employment, or doing work which amounts to them filling a role or providing short-term cover for a role within a UK based organisation. In addition, where the applicant is already paid and employed outside of the UK, they must remain so. Payment may only be allowed in specific circumstances set out in V 4.7.

IANAL and I’ve not been able to find a reliable interpretation of whether V4.6 ‘remaining paid and employed’ effectively allows nomad working by a visitor whose true primary purpose in visiting is tourism/leisure. It’s a gray area - see also Where can I travel for prolonged periods of time while legally working remotely for a company in my home country?