Will Apple replace the battery on an iPhone 5s? On Apple's service pricing page, it says you can replace the batteries on the following models of iPhone for £45:

iPhone SE
iPhone 6
iPhone 6 Plus
iPhone 6s
iPhone 6s Plus
iPhone 7
iPhone 7 Plus
iPhone 8
iPhone 8 Plus
and all other eligible models

Does the iPhone 5s fall under all other eligible models, or is it too old?
Although discontinued in March 2016, iPhone 5s is still supported for service and parts replacement by Apple (as of this writing).

Check with your local authorised service centre for availability.