Yes, if A and B are computable then A ∪ B is computable.

To decide if x ∈ A ∪ B, run u(x), defined as follows. 

  u(x):
  if x ∈ A or x ∈ B then
    answer yes
  else
    answer no
The tests for membership in A and B can be done because A and B are computable.