Be very careful with the mains connection! Really!
The transformer is most likely iron cored, heavy (2-3kg) and fit for 50-60Hz. In your UPS, the inverter is always working, either from this transformer, or from the 12V battery.
Continuity means a low resistance (typically <50 ohms). It's not zero, so it's not the same wire. It's a winding (coil).
You will have to deduct what goes where, since it's unlikely to find any info on your custom transformer.
Let's take an example transformer:
Note that V1 / V2 =almost= N1 / N2. This means that a transformer with 100 primary turns and 10 secondary has a 10:1 ratio that also applies on voltages. You give it 240V on the primary - it will output 24V. Everything is AC, of course.
So more turns means more voltage. But that wire has a resistance, so
the higher the resistance of the winding, the higher the voltage (assuming same gauge wire). And resistance you can measure without disassembling the transformer. Now we're getting somewhere - we can find out the ratio of every winding and have an idea of the ratio between the voltages.
You know that it's from a 12V powered UPS and that means 220V primary and 12-14V secondary. So you're expecting more or less a 15:1 ratio. On a simple transformer, that's all we need, but on yours there are be other windings.
An ideal transformer has no losses, so a 10:1 ratio means that to supply 24V at 10A you need to power the transformer from 240V, and it will only use 1A. Input power = output power, since there are no losses. Now, a real transformer has an efficacy of let's say 70% so values are lower, but the principle is the same.
Higher current needs thicker wires, so what we get from this is that
lower voltage windings tend to have higher gauge wires (relative to primary wires). This is a clear indication of which side is the secondary (thicker wires), since
transformers generally have all the primary wires on one side and all the secondary on the other. Even schematic diagrams are drawn this way.
OK, now we know which is the primary side and which is the secondary one. You have to measure the resistance of every winding and write it down on a diagram similar to this:
From your schematic, the left side seems to be the primary, but you have 2 windings here. If they are of equal resistance, then they are for 110-220V compatibility. They are wired in series for 220V, or in parallel for 110V.
However, I suspect that one winding is high R, meaning mains input, and one is low R, that the UPS uses to tweak the input, according to your mains voltage. Meaning if your mais voltage is too low, it "adds" some more turns to the primary to compensate for that. Even if it will probably provide some low voltage, I suggest you don't use it, as it's not very well isolated from the main 220V winding.
On the secondary side, you probably have the Red-Blue-White that is a high current (thick wires) center tap double 15V winding. Meaning 15V - 0 - 15V used for the powering the power inverter. Measure resistance from Red-Blue that should be equal to that from Blue-White and half of that from red to white.
You also have another secondary winding (Blue-Yellow), of thinner wires, isolated from the first one, probably for some other low-power circutry. It's resistance should be comparable to that of the 15-015 winding, since it is also low voltage.
I've drawn a schematic of what I thing the connection diagram is
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