Firstly, it should be noted that the Manchester encoding library which I am using receives messages in the background with the aid of a regular Timer 2 based interrupt. This should mean that I can receive readings even when the Arduino is receiving/processing/sending serial data.
No, it does not mean that. Serial data processing requires interrupts, too. It is possible that the interrupts are interfering with each other.
I am aware that Serial interrupts could interfere with my own interrupts. However, when I was using a Serial over USB connection this didn't seem to cause any issues.
This suggests to me that there is something about an active Bluetooth connection which is interfering with the reception of sensor readings.
You mean that one radio interferes with the other? Quite possible.
It is possible but is it normal/expected? I am assuming that the whole point of the circuit on my 315MHz RF RX module is to only receive data at 315MHz and to filter out other frequencies.