The shortest path may indeed change. This is not because of some property of the uniform cost search, but rather, the property of the graph itself. Note that adding a constant positive cost to each edge affects more severely the paths with more edges. Here is an example, where the shortest path has cost $5$:

Adding a cost of $1$ to each edge changes the shortest path in the graph as:

The original shortest path has a new cost of $10$, whereas the other path has a cost of only $9$. Therefore, any optimal shortest path algorithm, such as Dijkstra's or uniform cost search, will find a different shortest path.
The shortest path will not change if it also has the least number of paths among all paths from source to destination. If the shortest distance is $d_{min}$ and $k_1$ edges, and there is some other path with distance $d_{min} + \Delta$ with $k_2$ edges ($k_2 < k_1$), it can be shown that adding any $c > \frac{\Delta}{k_1 - k_2}$ to all the edges will change the shortest path, and therefore your result.
Scaling the edges by a constant positive factor (multiplying all edges by $c>0$) will not change the shortest path.