@Multichill, Jane023, Pigsonthewing, Charles Matthews: I suggest it might be worth thinking a bit about what's going to be the best way to go forwards with this, before jumping straight in to creating a catalogue on mix'n'match.
There are currently 212,000 paintings on Your Paintings -- this includes some by minor artists and in minor collections. We might ignore for example any by artists that have been marked Not Appropriate for Wikidata in the current Mix'n'match run. But it's still a very large number of items -- probably well over 150,000, since the minor artists tend to be represented by only a few canvases at most.
In contrast, I've run a few back-of-an-envelope searches at Wikidata:WikiProject UK and Ireland/paintings. Currently we have items for 4865 paintings marked as either located in the UK or from collections in the UK. This total is dominated by works by Turner in the Tate -- 3061 according to Autolist, added in bulk by Fæ (talk • contribs • logs). Some of these may not be oil paintings (I didn't filter by material used (P186) because coverage is not particularly brilliant); but in contrast Your Paintings only has 392 canvases by Turner.
Excluding the extra Turners takes the number of potential matches down to 2196.
That's still not an exact estimate, because on the one hand some of those items are not oil paintings, whereas YP only contains oil paintings; while on the other hand, there may be some further items that are in the YP list, but didn't make my count because they didn't have either a collection or a location set. (Though the number may not be huge, because in total we only have 2277 items marked as paintings with no collection set (Autolist) -- but I suppose there may be further items not even marked as paintings. I guess assessing that depends on how comprehensively the category system has been mined, to set P31=painting. @Multichill:, perhaps do you have a handle on that?)
Anyhow, the point I'm making is that using Mix'n'match to try to match from YP into Wikidata, we'd be looking at a hit rate of only about 1.5%, which is miserable.
Matching from Wikidata into YP is probably a better strategy, but even then some care is needed -- for example, here are 22 hits for "Bacchus and Aradne", which would rather swamp a title-only Mix'n'match search.
To be specific, the search really needs to include the collection and the artist, as well as title keywords. I'm not sure Mix'n'match can do that, out of the box. Collection+Artist seems to be possible at the Your Paintings site, eg example, but then it's not clear whether title keywords can be added. (We would also need to start populating BBC Your Paintings collection identifier (P1751) (Autolist) and BBC Your Paintings venue identifier (P1602) (Autolist) -- which are not equivalent, eg the various branches of the Tate).
It may be that harvesting the painting names + collections + artist, and then a bespoke search (or an adaptation of Mix'n'match search) has to be the way to go.
But I'm a bit nervous about that too, this time for political reasons. The PCF licence is pretty restrictive, that its data is not reusable, with only individual collections able to authorise re-use. Which is accompanied by an "everybody go away" robots.txt file. Plus there's no legislated "fair use" provision in the EU Database directive. So there's a limit to how much we should lift without permission. Matching the PCF identifier might be legitimately de minimis, but it's questionable how much we should assimilate beyond that. On the other hand, if we did get permission, there is more in the PCF data we could add, eg media and dimensions, plus the institution itself in its own database might have additional valuable fields. And with permission (but only with permission), there would be no objection to wholesale creation of Wikidata items which did not already exist.
So the first best next step forward might be to try to get permissions for data re-use from some of the most important of the galleries and museums -- eg the National Gallery London, and some of the others near the top of the list here.