My question isn't canon or proven, but it's based on some pretty common current understanding of technology, so it deals more with possibilities.
Walter isn't human. His memories and programming are probably stored in a very predictable, reliable, mapped-out fashion.
Just like if the heads on a non-SSD/Flash drive crashes, you can take it to a service that can dismantle the drive, remove the actual disc platters, read and restore any data that isn't on the part of a disc platter that is physically damaged. The data is actually stored in pretty random fashion, but since there are standards for where the meta-data for how the files' information is saved, they can reconstruct most of the files (again, if the directory information portion of the platters isn't physically damaged).
Programming has to get loaded onto the androids somehow. Also, if one goes bad, gets damaged, or otherwise needs to be replaced, would they just write off all the information gathered, or would they load it onto a new replacement model, to pick up where the other left off? I'd think that any kind of design planning would have had a way to onload and offload data, so there would be designed, built-in, normal means for both getting information off of one, and for another to take it on.
Once Walter is incapacitated, David has time and skills available, though it is not shown, to extract that information. As a souped-up computer-driven entity, himself, accessing that information in a more easily usable way (vs. putting it onto a laptop and then decoding files) might even be built into his own hardware, to a certain extent.