Lighted button that also lights when another button is pressed
If I understand your question correctly, you're using alternate action pushbutton switches, with the button lighted in one of the positions. If your switches are double throw, you just have to wire them differently. I'm sure you're familiar with what's called three-way wiring (in the US - two-way in the UK)for house lighting. Stairway lights commonly are wired that way with switches at the top and bottom of the stairs. Toggling either switch toggles the lights. The switches used are all single pole double throw, also known as form C contacts. You can find how to wire them in innumerable places online.