No real time limit.
For the next visa, you need to show 2 years worth of mail in both your names, so you have a few months of leeway. If you can’t get things in joint names, you’ll need a nice mixture of official mail in each name... so don’t change everything to your name or he won’t have any mail to show in his name.
The ‘some sort of number’ the GP sent you was your NHS number - it’s unique to you and is your identification number in the NHS system - it’s used to keep track of your medical records.
For the National Insurance number, some people get one if they will be making extra National Insurance contributions or need it for other things... but if you don’t plan to work, it’s probably not vital. You may need to show you are looking for work in order to get one.
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