What controls how quickly groundwater gets recharged from rainfall?
What controls how quickly they refill with rainwater?


Really it depends on where the aquifer is. Letting rain top off a Hawaiian aquifer may take a few weeks. However, refilling the aquifer beneath the Atacama desert may take tens of millions of years. That assumes that the recharge source is actually overlying the aquifer; in many cases, it's not, thus adding lateral distance that recharge must travel.

Regardless, drawing down an aquifer does lasting damage and they will never refill to the same reservoir capacity. Once groundwater is pumped out, the solid materials are then free to further compact and reduce the pore spaces that would be re-filled. In the case of California's San Joaquin Valley this sediment compaction, or, surface subsidence, has caused a permanent loss of nearly %10 of the aquifer's recharge volume.


  Could these reservoirs be refilled faster if an outside pump refilled
  them using rainwater?


In one sense, sure, there are groundwater recharge systems that work as closed-loop systems: groundwater is pulled out of the ground for use by a community, treated by the waste-water system after being used, then pumped back underground some distance upgradient. Aurora, Colorado, for one, has such a system in place. Note that the system isn't used to filter the used groundwater, but to keep it from being wasted when draining it to a local river. Furthermore, groundwater levels must be kept stable lest compaction ruin reservoir capacity.


  What are the energy economics on doing this for a place like Nebraska
  or Bangladesh?


They may be entirely manageable, but the politics would be NUTS! After all, rainwater is a "surface" resource that is promised to users downstream from where it falls. Diverting it underground for the exclusive use of a community would be like trucking parts of Himalayan glaciers to Delhi with no regard to downstream users who depend, for example, on the Ganges.

You could fill a couple of academic libraries with proper answers to your questions, by the way!