Flight Instructor with student reported a temporary loss of control due to wind gusts on short final resulted in them being aligned with the wrong runway.

Narrative

We were returning on a long instrument cross-country flight on an IFR flight plan following the commercial duties of PIC flight; on the ZZZZZ arrival being vectored by ATC over ZZZ1 towards ZZZ. Weather conditions were VFR at the field but gusty from the west. Following the hand off from Approach to Tower we were told to expect a landing on Runway XXL and to 'fly the numbers' as there was additional traffic approaching from the east for XXR to land behind us in a sort of criss-cross pattern (i.e. both of us coming to land for the far runway relative to our position). Approaching the field we encountered bumpy winds and an increased groundspeed around 90 kts. with flaps 10 degrees (nearing our light weight Va); with how nonstandard this flight was already I was nervous and shadowing my student's inputs on the controls. On short final we were hit with a gust of wind that rolled the aircraft to the right. I took the controls and applied power anticipating a go around when Tower called up and cleared us to land on XXR ahead of the aircraft. Now on a proper glidepath and crosswind corrections I landed the aircraft and was advised of a deviation with the controller remarking 'not sure if it's deserved' to contact Tower.Human performance considerations: This was a return flight from a very early departure leg that required a diversion following deteriorating weather conditions near our intended landing at ZZZ2. Following a stop for fuel we adjusted destination and picked up a flight plan in the air. This chain of events is extremely nonstandard for this route in flight training and the added stress clearly built for both student and instructor