I am gonna answer this in relation to the A330, which runs along the same kind of idea, except the Emer Gen is via the Green System instead of the Blue System. Slight Differences but nothin too major.
The Rat can Extend on 2 circumstances. There are 2 Solenoids on the RAT...one for Manual Extension via RAT MAN ON p/b overhead in cockpit Or 2nd solenoid via Automatic deployment Logic.
Ok, RAT drops for either of the above reasons...Airspeed causes RAT to turn, thus turning causing the Hydraulic Pump located on the back of the RAT to pressurize the Green System exclusively. The Green system then allows the CSM/G (EMER GEN) to run....i.e We now have Electrics provided. (limited as they are). Certain Shedding occurs obviously as the output of the EMER GEN is not the same capacity as the EDP's.
Coming in to Land, you are using the EMER GEN...under a certain airspeed the RAT will not be able to produce enough HYD Pressure to maintain the systems...this is when you can press the LAND Recovery P/B and this will then draw the power required via the Batteries.
You said in your post above " the 2 engines power the battery which powers the electric pump, ...right?"... The Engines are running...The Electrics that we have on the aircraft come from the Engine via the IDG's....basically,they convert the turning motion into electrical power...and this what the aircraft is powered from...under normal circumstances Engine 1 powers AC Bus 1 and Engine 2 powers AC bus 2. The batteries are fed from the DC Bat Bus. which under normal circumstances comes from DC BUS 1 via the TR from AC Bus 1. The Batteries are controlled by the BCL's...Battery Charge Limiters...these "connect" the batteries to the DC Bat bus whenever the voltage of the battery in question goes below a certain value...therfore the system now re-charges the batteries and when fully charged, disconnects them. The Elec Hydraulic pumps are not powered via the batteries. they are fed from the AC Bus.
Hope this helps!