This must depend upon the conditions in question, but I think it would not be very long.
The length of a generation for e coli can be 12 minutes or 24 hours, so that gives some idea of a typical time.
I did find this interesting case in the literature. They found that even as you subject e coli to minimal nutrients they are expressing genes that prepare them for a richer nutrient mix. The bacteria are primed for a flood of carbohydrates succinate or amino acids.
This study incubated the e coli overnight before taking their readings, so this puts an upper bound of about 12 hours. The implication is that if you are moving the bacteria to rich media it might be just a short time before they are in log phase growth.
In practice moving from LB plates to log phase growth is usually just like 4 hours. This paper interestingly shows that cold adapted E coli moving to other media react in a few hours, but can show behavioral changes days out.
I would say that 4 hours might be a typical expectation. 12 is typically used in the literature.