What will happen to a person after receiving a blood from a donor with colds during the donation?

Depends. Probably nothing if your immune system is strong and functioning and if you have immunity to the particular virus or bacteria. Otherwise you might suffer the symptoms of the infections.
Blood donor & colds. People with colds are often asked to defer their blood donation until symptoms disappear. This is not because their donated blood may transmit the cold virus to the recipient but to avoid transmitting a more virulent infectious agent they could have had that mimicked the cold symptoms (west nile virus, brucellosis, malaria, dengue). Rhinoviruses which cause colds do not reach the blood stream.