This page describes the general position in the UK and is not legal advice. Anyone selling trading software or trading for others should take their own.
What the FCA regulates
The FCA regulates specified activities: dealing in investments, arranging deals, managing investments, advising on them, and operating certain platforms. The line, called the perimeter, is around what is done, not what tool is used. A person trading their own account, by hand or by software, is not carrying on a regulated activity. A person managing someone else's account, or advising them what to trade, generally is, and needs authorisation.
Where an EA sits
Buying an EA and running it on your own account is you trading your own money. The EA is a tool, like a charting package. Selling an EA that the buyer installs and controls is generally selling software, not managing investments, because the buyer decides whether to run it, at what risk, and can stop it. That said, the way an EA is marketed falls under advertising and consumer law, and claims about returns that are misleading are actionable regardless of the perimeter. Selling access to your own trading through a copy or signal service, where subscribers' accounts follow yours, is closer to the line, and the platforms that host such services are the ones with the regulatory obligations. Running an EA on other people's accounts for them, under any arrangement, is managing investments and requires authorisation.
MiFID II and ESMA
In the EU, MiFID II sets the equivalent framework and the European Securities and Markets Authority, ESMA, sets product rules such as the leverage caps for retail CFDs. The UK kept those rules after leaving the EU, which is why the leverage limits on the leverage page apply. Brokers offering CFDs to UK residents need FCA authorisation; the register is public.
A developer sells an EA on the MQL5 Market to buyers who install it themselves. Not a regulated activity, though the sales page must not mislead. The same developer, asked by a buyer to "just run it on my account for me", accepts and takes a share of profits. That is managing investments without authorisation, and it is an offence. The software is identical. The activity is not.
For a buyer, running an EA on your own account at an FCA-authorised broker is legal, ordinary, and carries the protections that authorisation brings. Handing your account to anyone else, however the arrangement is described, removes those protections and usually breaks the law on their side. For a seller, describing the software honestly and never touching a buyer's account keeps the activity where it belongs.