In existing bathroom, we have one switch that turns on the vanity light and the bathroom exhaust fan and a second switch that turns on a heat lamp.

I would like to rewire so that the heat lamp switch would also turn on the exhaust fan. The simplest would be if I could run wires from the heat lamp to the fan, but would that then cause both lights to come on. I would like to keep the lights independent but that the fan would turn on with either switch.

If you can get a cable from the fan down to the switches, here you can use two double pole switches, each would use one pole to control the light and the other to control the fan.

This would eliminate the problem of both light coming on as you mentioned.