Talk to a hardwood installation company. You might be surprised at what they can do with your floor in your budget. It might be too expensive depending on the extent of the damage, but a lot of visual damage is actually acceptable in vintage hardwood floors, and they may be able to replace the bad pieces, resolve the squeak, and refinish the floor for a cost less than the overlay and new flooring would cost.
Otherwise, plan on shoring up the existing flooring as a subfloor:
Add new pieces of wood anywhere there's a gap of more than 3/8" in the floor. If the floor is very uneven with obvious depressions and wear spots, going through with a large floor sander can reduce the unevenness, and produce a level and flat enough floor. There are leveling compounds you can use at this stage, but if you do, choose one that is flexible and will not become brittle. Concrete based compounds are not suitable, as they will crack and pulverize, ultimately becoming ineffective over time.
Take the time to screw the wood to the joists below to reduce the friction noise (squeaking). This might take a significant number of screws and time. If the cause of a squeak is rubbing between two adjacent boards, a short screw in the crack between them can secure them together enough to eliminate the squeak.
Depending on what you want to place on top, add 1/4" luaun or thicker plywood - up to 3/4" underlayment - to resolve the remaining flatness and gap issues, and to meet the needs of the flooring you choose to place over the top. If you're going to put another hardwood floor down, you may not need thick underlayment. If you want to put carpet and pad down, you'll probably want thicker underlayment. Consult your flooring installer or the flooring instructions for the recommended underlayment type.
You may be able to get away without underlayment if you use a thick laminate flooring.
Wait it out. If this was your place, you might ponder dealing with the root cause (probably, as noted in the comment above, that they're too tight), but this will return to reasonably flat in the dry season. Delay is not a problem.
Best Answer
Most likely the nails that are working their way loose are smooth shanked and will continue to do this no matter how many times you "set" them.
This occurs due to ,not the nails moving, but the floor planks moving the nails. Changes in temperature, humidity, and vibration all exert forces that make minuscule adjustments to the planks and ultimately the nails.
If aesthetics allow you could either replace the wayward nails with a finish nail that has a more positive "lock" in the wood. Any ring shank or spiral shank type nail will hold more aggressively than a smooth shank. If you have access to the under side of the floor consider screwing up into the loose boards with an appropriate length screw.
If you'd rather not have counter sunk nail holes showing fill them with an epoxy type filler otherwise the standard wood putty filler will eventually break loose.