You may not even need a full wheelset.

If you've got 27" wheels, you can just get a 27" track rear wheel for $90-100, and then you can use your existing brakes. (You won't get bright shiny colors, but you can get a perfectly good wheel. They still make 27" tires, and 700c tubes fit just fine.)

The person who sells you the wheel(s) should be able to provide spacers so your wheel fits just fine and you won't have to reset the frame, and if you're capable of using a wrench, you can lock down the axle bolts tight enough that they won't go anywhere.

One issue you may face is getting a straight chainline, as your existing crank may not line up properly.

If it's not straight, you can either space the chainring with washers/spacers, or you can buy a new crank/bottom bracket that fits properly.

Good Luck!