Connect separate Heat / AC systems to a Nest E
To make this work you need a common 24VAC source for both units.  Chances are that either of your existing transformers will do the job but check the 24VAC requirements for each and make sure the capacity of the transformer you choose is adequate.  If neither is, get a larger one and use it.
You'll also need a C wire which is the &quot;neutral&quot; side of the transformer, wired to your thermostat.  Both units will need to share the C wire so you may need to do some wiring.
I drew the following sketch which I think should work for you using the connections you said were available on your two units.