Hi everyone. I currently have a Casio PX-870 and are wanting to upgrade. I have been looking at digital pianos in the £1000-1500 price range, the Yamaha P515 (with pedals and stand) and the Kawai CN29 are the two I have been considering. Which piano would be better? And are there any other pianos in that price range I should consider?

I would save up a bit longer and increase your budget. The models you are looking at are not a significant upgrade on what you have now.

I'm fine with it not being a significant upgrade, the main reason for my upgrade is due to the PX-870 having noisy keys (which I have had to have repaired twice) more so than a big improvement in action/sound.

The keys have quite different feel so I think you should try yourself. I liked the key action on the CN29 a lot (surprisingly, more than on the CA series), and the CN39 has better speakers. The ES8 is the slab version, but the furnitures tend to generally sound better at similar specs.
I didn't like Yamaha P and CLP keys as much; can't complain over N1X keys, but it isn't same league.
Also worth trying Roland FP90 / HP603, LX704... around this range. Casio GP300 costs some more. Plan enough time to spend in the store(s).

It's odd that your PX-870 has noisy keys, as mine are as quiet as a church mouse! And if there's a problem with yours being noisy, you'd think that Casio would either just fix it or replace it, no? Casio has been so fantastic with me; when my new PX-870 was shipped with a faulty sustain pedal, they couriered me a new one, no questions asked.

I am very tempted by the CN29, I use headphones so the speakers are not a huge deal. I have read that Kawai digital pianos can suffer from "clicking" key noise, is there any truth behind this? The last thing I would want is to replace my current piano for another piano that could potentially develop the same issue!

Casios developing noisy keys in use is a known problem, so there's nothing odd about it.

And they of course only fix it for free during the warranty period.

Agreed. I think in that price range the Casio isn't going to be any worse and might even be better in a lot of areas.
Just looking around, it would appear that key noise is a standard complaint about DPs of whatever brand.

I had a Yamaha P45 for over 2 years and never had any issue with key noise and that was half the price of the Casio. I've had 2 full key replacements in 8 months with the Casio.

Enter "Yamaha P45 key noise" into a search engine. Then do the same with every prospective model in your price range. I've had a PX-870 for over a year and have had zero problems. It shows that items that are mass-produced sometimes have problems.