If there is upgrade to Bitcoin Core, is it possible for the Bitcoin Core developers to make some changes mandatory for the non-Core users?
No.
Policy changes have no effect on the policy rules that other nodes may choose to follow. Consensus changes would be the only changes the could "require" other node software to follow suit, but even then, it's still up to the node operators to decide which consensus rules they want to follow by choosing which software they wish to run.