If i get bivalent covid booster and then came down with a virus later the same day, will my immune response to bivalent covid booster be less since immune system is also fighting a cold?

If what you think happens, there won't be recommendation for taking 2 or 3 combined vaccinations at the same time such as DPT or taking covid vaccine and flu vaccine at the same time. Body immunologic response to a cold virus is not much different from reaction to a flu vaccine.

Probably nothing at al, i.e. no effect on COVID booster effectiveness.