Hi Takayuki, although I had checked my PHP error logs previously, your comment about PHP error logs made me go back and double-check them to see if there were any clues. As it turns out there are two locations for PHP error logs on my server and I'd only looked in one of them, when I checked the other location I realised what the cause of the problem is thankfully!
Short answer:
Apache mod_security spam rules
Longer answer:
I recently did a lot of server updates and had installed the mod_security module for Apache, I had included a set of anti-spam rules and it was actually these that were intercepting some of the form submissions. The form in question here is actually used for people to request repeat prescription from a Doctor's practice and they were including names of drugs in the content of their message. So basically mod_security was matching up some of the drug names with its list of possible spam content and so the form processing was just failing silently and so the Contact Form 7 plugin just keeps sitting waiting for a response from the server. I have gone in and made some changes to my mod_security settings and the form now seems to be submitting happily so far!
Thanks a lot for your help!