A300 First Officer reported on departure experiencing multiple system failures and loss of cabin pressure resulting in returning to the departure airport.

Narrative

Right after takeoff we tried to retract the gear while in sys 1 landing gear position detector. The gear wouldn't go up so we switched to system 2 and the gear went up. Right after that; we got multiple failures. Pitch trim 1; ATS; CA Pitot Overheat; FO pitot overheat; stby pitot overheat; CA tat heat failure; cabin press reg 1; ADC 1. All the failures did not happen all at once. It started with the ATS and the loss of speed tape for the CA. Then we lost pitch trim 1. We leveled off at 8k and asked ATC for vector to stay around ZZZ while we were troubleshooting the issues. The CA switched to ADC system 2; fd sys 2. We used operating procedures about the issue directing crew to not turn off the pitot heat; even when ECAM is directing it. Then we lost cabin press reg 1. We completed the steps on the ECAM and Status page and cleared out to the memo page. We both agreed that the problem was probably some sort of weight on wheels switch or relay; resulting in the aircraft thinking it was still on the ground. At this point we had burned too much fuel to make it to ZZZ1; especially taking into account cold temps; possible icing conditions; and any potential unsafe landing gear issues with limited gas to troubleshoot. We told Dispatch we were turning back to ZZZ.We then [requested priority handling] and requested the ILS to XXR. Right after that the CA [and] FO; stby pitot overheat illuminated. We ignored the ECAM directing to turn it off; because of the operating procedures. We used the normal checklists and landed at ZZZ.Cause: Proximity detector processing unit failed.