The Deer Act provides for the hunting of deer and so regulates how, and it what manner, deer may be taken. As such, it also includes provisions for rehabilitation and/or euthanasia:
Deer Act 1991 Section 6, General exceptions:
(2) A person shall not be guilty of an offence under section 2 or section 3 above by reason of any act done for the purpose of preventing
the suffering of an injured or diseased deer.