Is there any issue with getting a haircut or cutting your nails on certain days like Tuesday?
I’ve often heard people in my community say that an Imam from the Ahlulbayt (as) forbade people from cutting hair and nails on Tuesday but without any proof. This has often led to arguments within our family aswell. Would be very grateful if you could please clear away any doubts.

No there is no prohibition at all during days and times of the week (except in hajj).
Yes there are times when it is makrooh (discouraged) but not haram.