No the lens equation 1/u + 1/v = 1/f is not useful in this circumstance, as the image distance v (labelled S2 in the diagram above) is unknown, and the only way to find that distance would be to cut the camera in half and measure it. Even if you did that (bearing in mind it changes as the lens focuses, so make sure you focus before cutting it in half) it's worth noting that the formula applies to single element lenses, in a multi-element lens v (S2) refers to the distance from the second principal plane to the sensor. There are various ways to determine the location of the second principal plane but you don't want to go there.
You can determine the 35mm equivalent focal length very easily by measuring the angle of view, but unless you know the sensor size you can't relate that back to the actual focal length.
If you knew the aperture you could obtain a rough estimate for the focal length by trying to measure the depth of field, but it you don't know the sensor size or f-number then you're out of luck.