Those 433 MHz modules use AM modulation, and when not in use the input to the transmitter must be driven to 0 V (to not output any RF energy). If it is not, interference may be the result.
For instance, if the digital port line is configured as input (default at powerup) and if there is a pull-up resistor, either on the Raspberry Pi PCB or internally in the chip, it may result in interference (depending on the value of the resistor).
This may be the case if the other Raspberry Pis are powered up, but the program or configuration command-line that change the configuration of the digital port to output has not been run yet.
A system based on AmForth on an Arduino Uno (actual supply voltage 5.2 V) was used, and the same transmitter module as in the question was used (with the same supply voltage). It could successfully turn on and off an RC-controlled 220 V relay (brand name "Perel", but by now likely out of production or sold under a different name).
The digital port on the microcontroller was then configured to input, so the input to the transmitter was floating. This did not result in interference.
Then a pull-up resistor of 10 kΩ was connected. This resulted in an intermediate voltage of 3.7 V to the transmitter, suggesting the input impedance of the transmitter module is on the order of 10 kΩ (24 kΩ for a resistive voltage divider).
It also resulted in interference. The transmitter was located 80 cm from the relay, and the manual control had to be brought within 30 cm of the relay to overcome interference. This fits with what was reported in the question.
With a multimeter, check the input voltage to the transmitter. It should read 0 V. If it reads 0 V, you can check if it is floating or driven by measuring to 5 V instead of GND - if it is not showing 5 V it is floating.