That's why I bought two incubators. I can do multiple hatches. I incubate them in one and the hatch them in another. The hatching one can be inexpensive and no need for a turner, I also have a mini bator (those ones that hold 7 eggs) so that I can hatch them when situations come up like yours. I have two eggs in there now. One that is pipping, and one that has about 4 more days. My broody hens let other hens lay their eggs with theirs and sometimes, even tho I mark them, I let a hen sit on them too long without picking her up and checking for new eggs. I just can't throw them out if I see any development in them, so I throw them in the mini bator.
I think any homemade incubator will end up costing you as much or more than just going ahead and buying one when you consider all the little parts you need to get.