You need to firstly cancel your old SIM card straightaway. If this has been lost or misplaced or damaged in any way you need to cancel it with your network supplier before even composing the letter.
Then all you need to do is to outline in detail your name and number and make sure you put a subject under the date. This will then enable the customer services team to know who they are exactly dealing with.
After this you should explain in detail you have cancelled your existing SIM card number and that you would like a new one to start straightaway. If for any reason you think someone might be starting to run up a bill on your old SIM card then again this is something you should explain and have it known that if you receive a bill for a vast amount that you will be contacting them to resolve the issue.
Alternatively, after you have sent a letter and you do not receive a swift reply you can always call up your network supplier in order for them to deal with your enquiry quicker. This may mean that you may be able to request a SIM card much quicker than simply sending a letter. Once you have received a reply or a response make sure you take note and file it away for your own records.