I may have found a solution! First, if you are having these problems, check the build of your iOS version. Go to Settings>General>About>Version. This should read 5.0.1 (9A405).
Upon closer inspection of search engines, I came accross a website titled "What is the difference between iOS 5.0.1 build 9A405 and 9A406?". As it transpires, there's a great deal of difference, for us 4S users anyway.
Here's the website (I hope I'm not breaking any rules by posting a link. Please let me know if I am!):
http://www.iphonefaq.org/archives/971716
The difference between these builds seems to be a fix for a problem in the iPhone 4S not recognising SIM cards. Apple's own knowledge base article states the following;
"In certain situations, iPhone 4S may have difficulty recognizing a micro-SIM card, resulting in the following behaviors:
"Invalid SIM" or "No SIM Card installed" alert appears intermittently.
Status bar displays "No Service" or "Searching" in a location with good network coverage."
Apparently, because this is a small fix for a small number of users, the fix has not been released as iOS 5.0.2. Therefore, people who updated to 5.0.1 early, such as myself, can't simply update by checking on iTunes or Software Update. In order to receive this newer build we have to restore the phone to factory settings, confirm the build has been updated to 9A406, then restore from a backup.
I haven't attempted this yet as it may take some time, and I don't have enough until later in the week, but it anyone else tries this beforehand and finds it works, please let us know!