Thanks to everyone who replied. 

I also posted the message to ESRI Forums (http://forums.arcgis.com/threads/23274-Converting-UKCP09-data-to-raster-format-creating-rotated-pole-custom-projection) so here is the summary of replies. 

The major problem seems to be with the Arctic Conical Projection, which is why the grid appears 'wonky' (figure1 and 2). This is, apparently, a common projection the the global Climate models and Regional Climate models where the UKCP09 data is generated from. 

One solution (suggested by a number of people) was to convert to raster at a much higher resolution than the 25km^2 grid squares. This would allow the data to be represented, but dramatically increase the resolution (and size) of the file (figure4). However, this would make my analysis more difficult as I want to use counts of raster cells, which would be difficult. Also, it appeared that when I did the conversion, the number of raster cells in a grid cell were not always the same, so I couldn't just scale the results back down. 

Another solution was to create a regular grid of points (vector), use the spatial join to get the values in the polygons, then convert that into raster, which I think would work. 

A rather 'thinking outside the box' solution was suggested, where ESRI Tech Support suggested rotating the polygon grid to a 'normal' horizontal/vertical grid. Of course the cells would no longer be in the correct spatial location, but this potentially wouldn't matter for the analysis. 

“You could convert the vector shapefile data from WGS1984 projection to British National Grid, then rotate the polygons using the Rotate tool. You would then get a grid that is perpendicular to North-South, that could then be converted to raster. However the resulting raster coordinates would not be British National Grid, because as you can see from the attached screenshot (figure5) the grids have moved towards the Atlantic. However they still of course represent the original 25 km grid squares on the ground.

The way I created the new shapefile is as follows: 1. In ArcCatalog use the Data Management, Projection and Transformations, Project tool to convert your WSG1984 shapefile using: OSGB_1936_To_WGS_1984_Petroleum transformation 2. Edit the resulting feature class in ArcMap 3. Select all the grid polygons so they are highlighted 4. Select Modify Feature as the Task on the Editor toolbar and the Target as the new feature class 5. Click the Rotate tool then press key A and enter an angle (I used 15 degree, but you might be able to calculate a more accurate value) However you could use the Spatial Adjust tools to move your other vector data to match your rotated grid.“

Another way of doing the rotation:

“I created some point features over the original un-rotated grid, then selected this feature class as the Target in the Editor toolbar and a task of Modify Feature. Using the Spatial Adjustment toolbar (under Editor > More Editing Tools) I digitised 4 Displacement Links from the corners of the original grid to those of the rotated one (the more links the better). I then selected Rubbersheet as the Adjustment Method, selected the points I wanted to move and clicked Adjust. I've attached a screenshot before (figure6) and after (figure7) the Adjust.”
I think the best solution (suggested by a couple of people) would be to create a custom projection in ArcMap, and with that allow the grid to be displayed horizontally/vertically. However, I don't have the knowledge or time to do this!

For my analysis, I am going to stick with the vector format, as I can do my analysis in this format, but it will just take a bit longer. All the potential suggestions had potential for introducing error into the analysis, which is something I would like to minimise. If anyone has any further suggestions or comments, then feel free to post to the list/forum or email me directly. 

Many thanks for everyones efforts,
Nick.