As a Swiss citizen I usually assume that I am not eligible for the funding schemes restricted to "UK/EU nationals". On day I still double-checked with one of them (Oxford Brookes University, in a wild hope), and have been confirmed that students from Switzerland were not eligible.

Now, for my future applications, do you think that Swiss nationals are in all cases never eligible for UK/EU funds, as a national policy?

Or do you think these policies are rather university- / funds-specific, and that it would be worth it to check individually for each PhD position I am interested in?

My guess would be it's an EU universal policy, see http://www.bbsrc.ac.uk/documents/studentship-eligibility-pdf/

EEA and Swiss nationals
13. Nationals of a European Economic Area (EEA) member state that is not part of the EU
such as Iceland, Liechtenstein and Norway, together with Swiss nationals are free to
live and work in the UK. However, they are not settled in the UK unless they have
applied for and been granted settled status, and are therefore not eligible for funding.
EEA and Swiss nationals who are resident in the UK may, however, be eligible for full
awards as “EEA migrant workers” (see paragraphs 18-21).

This only applies to funding from certain places though, so if funding is open worldwide you can apply.

Exactly the answer I needed! Thank you so much TreeofLife for your help!

Please bear in mind that anything that currently stands about eligibility may well be over-ruled at some point in the next 2-3 years. Nobody in the UK has a clue right now about how anything will work post-Brexit. You may well find in the future that you have exactly the same eligibility (or lack thereof) as EU citizens.