It depends on the credits. For the ones you mentioned, the retirement savers credit comes first (line 50), so is used first. There would be no tax left for the child tax credit, so that would be zero, but you should be eligible for the additional children tax credit, which is the refundable version. I have no idea why they have two versions of that credit, but you should get the refundable credit in the end.
Now, for other combinations, it could work differently. Personally, the retirement savers credit is worthless to us, we owe no tax. If we earned enough to owe tax to use the credit, we'd no longer be eligible.