I'm no expert but I would suggest that you need to do some research into
map projection techniques, and specifically, find out what projection your bitmap of the UK is based on.
Your conversion formulae are linear, which will only work if your bitmap is also 'linear' - ie if your bitmap is the same number of degress of longitude wide at the top and bottom. However, if that were the case for your bitmap, some distortion is inevitable - Scotland would be 'stretched' in the same way that Greenland is stretched in many atlases to appear the same size as Africa, when in fact it is many times smaller...
If your bitmap isn't linear in the sense described above, then you'll need non-linear formulae to do the mapping...
Good luck!
cbl