Finetuning is a relatively quick process. Training the base model is the expensive part (can take weeks and huge amounts of compute), whereas finetuning usually is only on the last few layers and can be done with much less resources. You could definitely have a "nightly" finetune model that is retrained every day or so.
Interesting - how would that work for a company that wanted to run their own codex model, on-prem, trained on their own code? Perhaps also trained on their dependencies?