This is preliminary ...
The low resistances measured on winding 1 may be due to a damaged transformer, but they may also be due to a bad power supply capacitor.
To check the transformer by itself, you must measure the resistance of each winding. Winding 1, the high voltage secondary, is the only one which is grounded - 1T is the center tap. (1S is the start of the winding, 1F is the finish.) In addition to measuring resistance from start to finish rather than from terminal to ground, you must disconnect other things attached to the winding to get accurate values. That means unsoldering the wires connected to 1S and to 1F. Measure 1S to 1T separately from 1T to 1F.
Windings 4 and 5 appear to be the 2.5v windings. They are paralleled to drive a 2A3, so you'll have to disconnect at least one of the terminals and remove the 2A3 to test each winding. Both wil be ver low resistance, less than the 0.5 ohms or so residual resistance of your meter - you are just checking them for whether they have opened up (infinite resistance) or not.
That leaves windings 2 and 3. I'm not sure which is the primary and which is the driver heater power, but you just have to remove the driver tube to measure each of them. The primary will be about 5 ohms, the other will be very low.
If the transformer is OK - and it may well be OK, we don't know yet - then you'll have to look for short circuits elsewhere to explain the smoke. In fact, even if the transformer is not OK, it may have been some other short circuit that overloaded it. As Doc said, the power supply filer capacitors are the first place to check.
Since this amp has the Tucker board, that's another thing to check. Because it's direct coupled, a failure of the driver circuit could cause the 2A3 grid to go positive, up to the power supply voltage; that would cause the 2A3 to conduct too much current, overheating the transformer and other components and causing the failure. In that case the cathode bypass capacitor (220uF, 250v?) may also be toasted.
