I guess the OP has sorted this out by now,but,if he has found the two terminals which are connected with the switch in the "on" position,he knows that pair of terminals need to be in series with the Active or "hot" side of the Mains incoming,& one side of the transformer primary.
The other side of the primary goes to the Neutral side of the Mains.
With the DMM,it should be easy to determine the resistance of the transformer primary.
(It may be quite low,but it won't look like a short circuit)
Wire up the switch as suggested above,then with the switch "on" check the resistance between Active & Neutral
of the device's power plug.
If it looks the same as the primary direct,it is correctly wired--if it looks like a short circuit,rewire it,so it is correct.
Ok,so far,so good,but here is where the fun comes in..........
The third (unswitched) terminal should not have readable continuity to any other terminal at any time.
(You can't see the Neon with your DMM,as it needs enough volts to ionise the gas inside it.)
To make the Neon work,you need to connect that third terminal to Mains Neutral.
The problem here is that if the terminal coming from the Mains Active & that going to one side of the transformer primary are swapped over,all will still be safe,but as Monkeh pointed out,the Neon will always be on.
If so,swap those two connections over,& everything will work as required.
Using the resistance range of a DMM,everything can be wired up safely without having to connect the Mains.
A Mains connection is only need for a brief test to see if the Neon works as required.
After the resistance testing,return the DMM to "AC Volts" or "OFF"..