yeah my first thought was to recommend function of beauty...I don't know if they have a UK base...they allow for making custom hair goals and such (you can focus yours on dry hair). It has done wonders for my oily hair and not hurt the dye. My mom has really dry hair and after a month of use it really helped her. The other thing we did that helped her immensely is hot oil.
Choosing an oil...is not as easy as it looks. I would research what oil(s) fit what you need. In dry hair sometimes it has trouble sinking in and staying, so heating the oil can really help. This is somewhat difficult since you won't need a good deal of it. I heated my mom's oil by first heating hot water in a pan, then putting a bowl with the amount on top of the water. this heated the oil for use. Then you apply it all over the hair, make sure you're getting toward the roots, and leave it covered and in for 30 minutes. My mom bought a shower cap for it because otherwise the oil fades too quickly.
Mielle Organics Rosemary Mint Growth Oil | Walgreens
this is the exact oil she used to help. I am not sure it'll help but thought I'd show because in the photos they have ingredients too so you can maybe find something similar?
I hope this helped a little.

dry hair can be so irritating.
When I get hair dye dryness, I tend to find deep conditioning to be helpful. Some stores will sell packets of conditioners with similar oils to above. I personally recommend argan oils and sunflower oils, those always seem to help my hair. I find coconut oil softens it quite a bit too. The main one I have had infuses sunflower, avocado, and argan oil. I would recommend leaving this in for like 30ish minutes and then rinsing it out. There's other store packet type stuff and you can find what is right for you. I find drugstores have more of a selection on this here, not sure how the UK works.