Great question — answering as the UphillTech team:
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...
Full answer by Willeke: https://outdoors.stackexchange.com/a/28741 (CC BY-SA 4.0).
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