Maybe Oracle is working on setting up a new base in Gotham, since it does seem she's decided to move back. But Steph clearly wants to crimefight now, and the batcomputers are the only ones set up right now to hande Oracle operations, and no one's using them. Heck, did we even see Bruce's HoloWill to Babs? Maybe he specificly gave the batcomputers to Babs, since she was the one who could do the most good with them.
And as for wehter or not they're welcome, Bruce is gone, and Dick and Alfred likely have no problem with it if Oracle wants to set up shop there. Especially since IIRC Dick decided to base himself elsewhere due to the emotional ties the Batcave has for him.