Never mind, I've fixed my issue.
gwforeman's suggestion was right, some dirt was indeed trapped right at the top of the lightning cable socket, preventing the cable from going all the way in. It seems when I plugged it in previously it must have compounded the dirt and stuck it right in the top which was very difficult to see, even when shining a torch in it.
I used a pin to gently scrape out the dirt, now the cable plugs in without a problem and the phone is charging again.
Peter Gillespie, your suggestion would have been fine if there was an Apple retail store with a genius bar anywhere near where I live. Sadly, the North of England is forgotten about as usual and my nearest store is over 60 miles away making it not very feasible (especially when I don't drive)! I often see people replying telling others to go to their genius bar, but please remember that it isn't always a possible option for many people hence why they're posting on these forums (but I appreciate your quick response nonetheless, don't get me wrong!).