I don't think anyone in the modern RAF will know much about astro navigation, and this is such an obscure corner of RAF history that I doubt the museum will know much either . But if you want to know more - here goes, this should kill an hour or two:
Astro navigation is based around measuring the elevation of a heavenly body, ie the angle between it and the horizon. On ships this is easy enough to do with an optical arrangement that superimposes the body on the horizon and you read off the angles. In an aircraft at night finding the horizon is not so easy so sextants were devised with built in ways of defining the horizon(tal), This was done with a spirit level like arrangement called a bubble sextant or a more modern one with a "pendulous reference". In each the operator kept the instrument level using the reference and "shot" or measured the elevation of the body. In an effort to cancel out errors these instruments did not take an instantaneous measurement but took 60 shots over a minute and averaged out the result using a clockwork device.
During the minute "shot" the aircraft obviously has to be stable. Any accelerations will act on the bubble or the pendulum to give a false horizontal and thus an erroneous elevation of the body. Fore and aft accelerations manifest themselves as speed changes during the shot and left and right errors as heading changes.
This is what the Mears Slide was designed to compensate for. You could set it with the speed and heading changes and it would tell you corrections to be applied to the measured elevation.
To plot an astro shot you had to assume a position where you would be at the required time. It would be the nearest whole number of degrees of lat and long. You then entered a lot of tables in thick books and extracted the elevation and azimuth (direction) of the body if you were at that assumed position at the calculated time. When you took the shot the difference between the calculated elevation and the measured one was called an intercept and plotted as a position line on the map as nautical miles towards or away from the assumed position (in the direction of the azimuth). The Mears Slide corrections were applied to the intercept towards or away from the body.
That would all have been easier in front of a blackboard but I hope it makes sense.
Now that is out of my brain perhaps there will be room for how to work my Christmas I Pod!