In the scenario where I'm living in the UK and working remotely for a Swiss company. Who should I pay income tax to?
I would be declared self employed in the UK.
To HMRC. You pay tax in your tax domicile (i.e. where you more or less permanently reside) – in this case, the UK.
Rules are, of course, a little more nuanced than that. For example, if you don't bring the money back into the UK at all (e.g. if you get paid by your Swiss employer into a Swiss bank account), the situation may change.
The guidelines for whether your foreign income doesn’t need to be subject to UK tax are here: