We humans have mostly frozen weights (neurons), or else we would constantly be having to avoid forgetting how to walk+talk. We have a period of greater plasticity (youth!), and use sleep and dreams to perform 'deeper' updates than occur when we're awake: We tend to suck a bit at picking up new skills from zero, but improve rapidly with practice over days.
Input stream comes in, input stream comes out. The LLM doesn't care whether this happens once a minute or once a year.
Nothing prevents one from running an LLM whose harness has a clock and a while loop in it, and it would be weird if its mere lack were really so consequential to consciousness.
Your brain and your whole body exist in time. Even when you are asleep, your body does not flicker out of existence and your brain actually continues working during that time.