Have I erred in performing the prayer after waiting and the time for it has begun? Is it permissible for a worshipper to prolong their prayer after the congregational prayer has begun, or must they shorten it to catch up with the congregation? And is it permissible for them to act by re-establishing the call to prayer (Iqamah)?

If the regular Imam is absent, and delaying the prayer causes hardship for the congregants and travelers, then there is no blame on the questioner for establishing the prayer without the regular Imam's permission. The Imam must be gentle with people and not cause them hardship. If the prayer is established without his permission, it is considered valid, and there is no need for a new establishment.