Air carrier Captain reported a ground conflict during push-back with another air carrier aircraft pushing back at the same time.

Narrative

We departed ZZZ1 late after maintenance delays for GPS and TCAS required deferrals. Upon arrival at ZZZ; our gate was changed from XX to XY. We pulled into XY uneventful. Outbound flight in this aircraft was a quick turn due to the delay. After conducting the Parking. Checklist; we were packing up the cockpit and noticed the jet bridge had not moved. Jet bridge operator told us through the open window that the jet bridge had no power and this had happened earlier that same day but it was repaired. Within 7 minutes of parking; we noticed fuel level indicators showed fueling in progress. We directed ops on radio to immediately stop fueling as the main cabin door was never opened and we had all pax onboard. Fueling did not stop for another 2 minutes. Clearly a procedural and communication problem at this station. Bags were also being unloaded during this time.Once the tug; towbar and crew were assembled to move us to another gate; we requested and were given clearance from Ramp Control on the radio to pushback and reposition. During pushback to lane XX; our tug driver began frantically telling us that Aircraft Y was being pushed back quickly and was going to hit us; so he reversed direction and pulled us back toward gate. Ramp Control then told us we were clear and there was enough room for both aircraft clearance by at least 25 feet. No guidance given from ramp control to either aircraft pushing that we would be pushing into each other.After Aircraft Y was stable on lane XY; our Pushback and reposition to XZ [Gate] proceeded again with no further issue.