Python 2, 84 8175 bytes
lambda c,a:reduce(lambda t(h,bm),(d,e):all(bd&e)and(0,0)or((t[0]+b[0]h+d)%24,(t[1]+b[1]m+e)%60),a,c)
EDIT: Saved 3 bytes by replacing b[0]&b[1] with all(b); thx 2 Eric The Outgolfer.Try it online!
Function that takes time as a tuple (hour,minute); outputs same way.