Transformer Problem Diagnosis
It is unlikely that the red wires actually have no continuity if they are really the primary winding. If that was the case, the transformer would do nothing when connected to the mains circuit. If you are using a multimeter to check for continuity, it is trying to measure impedance and give you a value in ohms. I have found that cheaper multimeters seem to get confused by the high inductance of the primary circuit of small transformers. I have known-good transformers that read no continuity with my cheaper multimeters. 

An alternate test for checking continuity is to wire the primary in series with a circuit that has a DC current. For example, a 5 VDC power supply, a 100 ohm resistor, and a cheap LED. If the LED lights, the circuit has continuity. The LED must connected in the correct direction to be forward biased. This is a cheap, effective circuit for checking DC continuity. The peak current is only 50 milliamperes, so it is unlikely to cause damage to the transformers winding. The LED will be a bit unhappy at this current level but will not die instantly.

If you have a transformer that is tripping an RCD (also called as a GFCI), there are a few possibilities


You have connected it wrong. The circuit theoretically can work using the line and ground wires. This is unsafe however, this trips the RCD. It's also worth mentioning the outlet could be wired wrong. However, in that case anything plugged into that outlet should trip it.
The transformer is shorted to its metal case. If the transformer has its 220 volt primary shorted to its metal case (even with a very high impedance) it is allowing current to flow from the line to the ground. This trips the RCD. This is very dangerous if it is true. You can test for this by placing the transformer on a very good insulator, like a sheet of plastic. If you have a kitchen cutting board that is plastic and white in color, that could be used. If it does not trip the RCD when placed on an insulator, it is likely that it is shorted in someway. Do not touch the transformer whatsoever while performing this test.
You have an RCD that is also an arc fault detector. Some RCD breakers have a circuit to detect the energy from an electrical arc and trip. It is possible that due to poor manufacture or lightning damage, there is a tiny arc in the primary of the transformer. This is usually the result of several turns being shorted together by a carbon track from a lightning strike.