If all edge weights are 1, then any spanning tree is a minimum spanning tree. Thus we can forget the fact that we want a minimum spanning tree and only focus on spanning trees.
This means that using prim's algorithm to construct the original tree is a waste. Prim's algorithm requires that you add the least weight at each step. But since all the weights are the same it doesn't matter what edge you add as long as the edge doesn't cause a cycle.
If we remove a node, there are two possibilities. If the node was a leaf on the tree, the new tree will still be spanning the entire graph. In that case we are done. Otherwise we end up with two trees. It could be that the graph is now disconnected in which no spanning tree exists. Assuming that the graph is connected, then adding any edge to the spanning tree which connects two trees will work.