How does the radio altimeter reading change with the bank angle?
The Rad Alt will show the same value of 1000'.

The radio altimeter antennas provide a radiation pattern that covers at least +/- 15 deg in pitch and +/- 20 deg in roll, though most cover more - typically +/- 20 deg in pitch and +/- 30 deg in roll.  

The transmitted signal fans out equally over the entire pattern area from the transmit antenna and is reflected back where it is received by the receive antenna. The returned signal is spread over a short period of time due to the varying distances the signal must cover. The closest point will be the first received. As the aircraft changes pitch or roll, the location of the closest point will move within the pattern, but the shortest distance will still be the same.  

The DSP software also processes the signal to filter out smaller objects such as the approach lighting towers.