When searching in a tree, my understanding of uniform cost search is that for a given node A, having child nodes B,C,D with associated costs of (10, 5, 7), my algorithm will cho
Nope. Your understanding isn't quite right.
The next node to be visited in case of uniform-cost-search would be D, as that has the lowest total cost from the root (7, as opposed to 40+5=45).
Greedy Search doesn't go back up the tree - it picks the lowest value and commits to that. Uniform-Cost will pick the lowest total cost from the entire tree.