I plan to go on a hike with someone allergic to latex.
Usually I would pack small plasters and leukotape, but this is not an option in this case.
How can I replace these items with latex-free equivalents in the UK?
In short, all items containing latex in most first aid kits can be replaced by latex free products, a well stocked chemist or drugstore. And with a bit of preparation time they can order what is not in the shop.
For plasters you have found an alternative from St John, you mentioned in a comment, there are (other) brands found in an online search, so you have the choice of several.
For sticky tape I found several looking for 'latex free medical tape', I guess you will find more if you also search on sports tape and plasters (as that gave good results in a Dutch search.)
As indicated in the comments, latex free gloves are almost standard in the more expensive kits, even when not stated, but buying a box of them should be not too hard.
And think a bit out of the box, if medical sticky tape is hard to find, you can often also use a 'wind around the hand/arm/leg bandage' which you fix with an 'on the bandage' set of hooks on elastic. Or even with a bit of non medical tape if it is acceptable for this person.