It is incorrect to assume that every act begun at a prohibited time must be completed simply because it has been started.
For fasting on Eid, the forbidden day is inseparable from the specified fast, so formation of a vow does not license persistence in the prohibited act.
Prayer in certain disapproved times has its own detailed rules because the role of the time differs from fasting. Performance later may be possible.
Principle: the act’s basic legal category, the vow’s formation, beginning during a prohibited period, and the requirements of completion or replacement are separate questions.