[adw]

It is common for big workplaces to have multiple unions and essentially all unions are sectoral and role-specific rather than company specific.

Take the NHS; it will have to deal with ten plus separate unions - https://nhsunions.org/#about – of which the biggest powers are the British Medical Association and the Royal College of Nursing, but the cleaners are GMB or Unite and they're huge pan-working-class institutions.

Education has to deal with the NEU, the NASUWT, and the NAHT, each of which has a different political slant. Some unions in the UK have been historically rather centrist in their politics (a good example of that is Prospect, https://prospect.org.uk/about/, which is a roll-up of a number of scientific and finance unions), some are firebreathing communists, but all of them work across employers.

There's also no such thing as a closed shop in the UK – because there are much stronger worker protections, there's less of a need for one.

(I was, at one time, in a majority-UCU workplace; https://www.ucu.org.uk.)

[RobotToaster]

There's also nothing stopping people being in multiple unions, a lot of IWW members are.