I will just run through the hydraulics as lots to read here.
As you say each engine has a mechanical driven hyd pump - four systems.
Each system has an bleed air powered demand pump (ADP) in the rear of the pylon, also in each pylon is its own hydraulic system reservoir.
No.4 pylon also contains an AC powered aux pump (ACP) for primary brakes when on the ground.
As an extra some aircraft have an AC aux pump (ACP) in No.1 pylon for steering during towing.
Confusion:
On later aircraft some demand pumps changed from pneumatic to AC powered.
You will see the Nos 1 and 4 with bleed air and the nos 2 and 3 with AC powered demand pumps, but was dependant upon customer choice.
APU does not have a hyd pump fitted.
The APU will power the demand pumps with both air and AC electrics as required on the ground.
If an engine driven hyd pump fails or hyd pressure drops too low its demand pump will automatically cut in to supply oil flow and pressure. This pump will be powered by engine bleed air or AC electrics depending upon position.
If all 4 engines were malfuctioning that would be another story.
But I take it this was all 4 engine driven pumps then the demand pumps would auto cut in as required and then stop when the need had finished, the power would come from the engine bleed air or AC electric system.
Nos 1 and 4 are the most used hyd systems especailly during initial climb when gear retraction occurs and t/edge flaps are stowed and then similar for descent.
That's the basic layout but there will always be an airline that had some differences built in but the above is much the norm.
TW