Depends on the manufacturer and what policy they have in the regional market (UK/EU in this case).
You have no statutory rights for anything bought outside the EU (and after Brexit, probably anything bought outside the UK). It’s purely down to the manufacturer then.
For example, though not bike related, my Nikon SLR only has an EU wide warranty, but Nikon have a worldwide warranty on lenses so any lens I buy anywhere in the world is covered if I send it to any local Nikon repair.
Also, you may not even get warranty sending it back to the US. It depends on their policy again. Problem is you exported it and assuming you’re not a US citizen you may not get any warranty. But yes, if you did, you’d have to pay the postage. Same though can be true in the UK.