Whether your union is a union having an intersection or one-pointed union (wedge sum) or disjoint union, the fundamental group of your space is trivial, implying that every loop in your space is homotopic to the constant map.
For example, if your union is one-pointed union (wedge sum) and you apply the Van Kampen theorem, the fundamental group of your space is still trivial since the free product of two trivial groups is trivial.