One method: If all the images are taken from the same viewpoint and at the same scale, and if you knew the scale of the image (i.e, 1 px = 10 arcseconds; you can work that out from the physical parameters of your scope and your camera), you can then put the coordinates $x$ and $y$ - in pixels - of each star image into the general equation of a circle:
$$(x–h)^2 + (y-k)^2 = r^2$$
then, solve the resultant simultaneous equations to give you the parameters (center (h,y) and radius r) of the circle the star traces around the celestial north pole. Convert that pixel value to a angle value using the aforementioned scale (for example 1px = 10 arcseconds). Now you have the declination.
It might be simpler to calculate the coords in terms or alt/az, so to get az, you can use the pixel x coordinates of each star, and subtract that from the pixel x coordinate of the center of the circle you found. Then, you can do the same for the y coordinate. Scaled to an angle, the azimuth number can be used unadjusted, but remember that the center of your circle will be positioned at 0 degrees from North, but angled from the ground by (90-your latitude) degrees. You can then scale the pixel y distance from the center of the circle to get your angle, but then remember to add (90-your latitude) to get the angle relative to the ground.
Then, of course, for each alt/az value, you can use LST and your location to convert to a RA/Dec value if you so require.
I imagine this whole process wouldn't be too difficult to automate; the great thing about astronomy is that our datapoints - images of stars - are very, very easy for a computer to read. You'd just need to feed each image one by one - timestamped with LST - into a program to find the coordinates of the bright spots against the totally black background, then carry out the aforementioned process to produce an epoch-referenced RA/Dec position.