Well just to give you all an update, I had another crack last night at re-soldering the pickup and even though I believe there's a marginal improvement, I still don't think it's 100% :(
Unfortunately, my iron (which although 30W was part of one of these electricians kits from B&Q) isn't the best quality and the flat headed tip now looks more like a fork - haven't a clue how that happened but am hoping that it's a poor quality component.
I'll try and get a replacement iron tip today but am not feeling very confident about successfully resolving this. I'd very much like to be able to suss this out but it's looking more likely that I'll have to take it to a man that knows his onions!
Am now worried that if I keep hacking away that I'll end up creating a bigger job than when I started which is ultimately going to cost more to resolve when I concede defeat and take it to an expert.
Most frustratingly of all is that even though I tried to adhere to your advice about heating the component and allowing the solder to flow onto it rather than the iron tip, I could not get this to happen. I'm hoping this is because the tip was screwed but it's also quite possible that soldering is very low on my list of strengths!!!
All very depressing really! :(