London to Andalusia by train?

Since when? Last year they were still really bike friendly on Eurostar, at least if you booked ahead - Brussels to London was a breeze.

You can take bikes on Eurostar but have to book then with EuroDespatch? first. Read the Eurostar website.

TLDR for Renfe: Check very carefully the dimensions of your packed bike before you get there; book your train tickets in advance and try to verify them close to travel date; build some margins in to your schedule; also avoid queues and disappointment by booking tickets to the Alhambra in advance (the deal where you go to the Nasrid palace, which is the most popular part, in the evening and then the tickets were still valid the next day to visit the rest is pretty good because the palace is just as spectacular by night).

We took bikes by train from the UK to Extremadura in 2014, and have also been by train from the south of France to Andalusia without bikes. I would love to tour in Spain again.

Eurostar I think is easy enough to find out about. If you can drop off your bikes the day before you avoid the problem with them possibly going on a later train to you. You can cycle across Paris, but then IIRC you have to put them in a bag of no particular size for the TGV.

Renfe, once you're on the train is amazing (they display the speed in the carriage and it goes incredibly fast but is completely smooth, it's spacious, it has "in-flight" movies...). But, we needed to pick up our tickets in Barcelona and that part of the station was pandemonium, so under time pressure at the airport style security/check in we found not only that bikes had to be in bags, but that the sizes they allow are specific and really quite small and they are quite strict about it. We had very cheap bike bags that were not rigid and with a bit of pulling them about we got the dimensions close enough to persuade them (my wife speaks basic but enthusiastic Spanish), but even after the dash from the ticket office it did look for a few minutes like we might still get stuck in Barcelona. After Madrid we must have travelled by local trains, on which you can conveniently transport your bike whole, but to get to Andalusia you would probably want another high speed one. Navigating the main Madrid train stations doesn't seem trivial to me, but when we had the bikes with us we had plenty of time for that. In Extremadura we went on a week long tour in a loop and just asked if we could leave the bike bags at the hotel we stayed at on the first night, this worked fine.

I had another strange experience with Renfe on a different trip when I bought a ticket at a machine with my credit card, presented it at the gate not long after to board the train and found it had been cancelled without notice (apparently by the card company) and I had to buy another and wait probably 2 hours for the next train, with no guarantee it wouldn't happen again.