How will axial precession affect the weather in the northern hemisphere (UK latitudes)?
You may be better off getting a more precise answer by cross-posting this physics if you want a precise mathematical explanation of the model. 

First, axial precession, as mentioned above, is different than axial tilt. Both change over time, but on different time scales. 

That said, the answer is it has most certainly been explored: Milankovitch cycles are an attempt to explain patterns of solar forcing on climate. The wikipedia article does a fairly good job of laying out the foundation of the theory, which is that there are three key components influencing solar forcing: eccentricity (how elliptical our orbit is), axial tilt (the change in amount of tilt of the earth), and axial precession (the actual change of direction of the poles). All three components operate on different time cycles.

Unfortunately for us, the nice Milankovitch theory leads to a huge conundrum: the predictions don't play out as expected. Physics dictates that eccentricity would have the weakest effect of the three, but the largest fluctuations are seen in roughly 100ky intervals (tilt and precession operate on roughly 25k and 50k intervals, eccentricity roughly 100k). This leads to the 100,000 year problem. 

Now, all of this is in the context of climate on the global scale, but the point applies just as well: nobody has a conclusive answer as to what exactly is going on, hence the entire field of paleoclimate. Unfortunately for scientists, the Earth is far from a simple rock: our physical (ocean, atmosphere) and biological systems are so complex that it is often too simplistic to pin down changes on one particular phenomenon on a geological time scale. People can study the temperatures  of individual sites over time, but determining the big picture cause of those fluctuations is tough enough for the planet as a whole, much less for a location as specific as the Salisbury Plain.