Again, many thanks for all comments. Really helpful!
Bit of an update: I had a go in an obscure corner with white spirit+wire wool - it sort of worked, but it was hard going and didn't end up with bare wood. Looked like being pretty hard work too, given the areas I'm looking at. I ended up sending an email to Fiddes, who make the
waxoil that was originally used and is now the issue, in the reddish areas, and got a swift response from their Chief Technical Officer asking for my number. (Excellent customer service, Grommet!) Over the course of a chat he persuaded me that waxoil being as tough as it is, removing it entirely would indeed require more than wire wool. So. I think I'm for some sanding.
Various comments upthread have only reinforced my gut instinct that a full-on hire sander would be OTT/overly-aggressive for my needs, and likely to be more trouble than it's worth. Also bearing in mind that this is indeed engineered timber, albeit with a 7mm top layer, so I don't want to go taking mm's off willy-nilly. Also, instinct tells me hire-shop jobbies are great for doing large unrestricted areas, and mightn't be at their best in situations like mine, where almost everywhere I want to sand is an edge/otherwise fiddly. I'm thinking my usual B&D hand sander (non-rotary) with medium/fine sheets.
The ultimate finish, incidentally, is slightly up in the air. Personally I'm minded to take it back to bare timber and seal it. Period, as they say. I actually like the natural greyness. If you've got oak, let it be oak, what? 'Er indores prefers a stained finish - if she had her druthers it would be reapply the mahogany, or whatever it was we used. We've been prevaricating for years on this sticking point, but I think I've finally worn down her resistance and she'd go with pretty much anything so long as it gets done.
Anyway, thanks again for thoughts - very helpful. I'll post back when I get round to doing it.