No.

What if they have a Welsh passport?

Around 25 million non-EU citizens live and work in the EU. So the answer is Yes. Do you have the right to go and live in the EU for months on end and take a job as a barista, or some other job that a local could do? No. But you are absolutely allowed to work in the EU as long as someone wants to employ you. Being a native English speaker you have a pretty good natural advantage -- and I don't mean that in a flag-waving sense, obviously. But most international business these days is conducted in English so there are some openings that you'll be more suited to than locals who don't speak English.

No based on their Swiss permit though, which was what the OP was asking. Only by going through the normal Non-EU processes.

I thought he was an 'Englander'? But in any case, if he/she can still go through 'non-EU processes' if offered a job then what's the problem? The fact remains that they can work in the EU. It's one of the great myths of Brexit that, OMG, no one is allowed to work in the EU anymore. Of course they can. Brits can work in the EU the same way that Brits can work in other non-EU countries, or the way that other non-EU people can work in the EU. It's no longer a 'right', that's true. But it's perfectly possible to do so.

Of course a British person can work anywhere in the world if they go through the correct process and get approval. But that was not the question! The OP wanted to know if his Swiss permit gave him working rights in the EU.

You use the words "can" and "able to" - you're perfectly correct. However the OP used the word "entitled" - to which the answer is a big fat NO. UK citizens can work in the EU but are no longer entitled to - it's part of the definition of Brexit. And a Swiss permit has absolutely zero bearing on work entitlements in the EU, any more than EU permits have any bearing on work entitlements in CH.