The ones from the meter to the CU are yours, but the meter design may or may not allow you to replace them without breaking a seal.
And if you don't have an isolator somewhere you'll need to break the seals on the cutout to remove the fuse. NOT A DIY JOB.
Also in England & Wales it's notifiable, the DNO may impose special requirements depending on the new distance between meter and CU, and because of all the circuit cables moving a CU is non-trivial.
Technically, the answer is yes, although the electrician would have to work live. There's no need to remove the existing tails from the meter unless you want to upgrade them. Remove them from the CU and put them in a Henley block (or, better still, an isolator) and then continue from there with new tails. For what it's worth, I would say that cutting a live tail outside the confines of the CU is safer than trying to extract one intact.
There have been a number of posts on this forum about the relative merits - and dangers - of working live or pulling fuses, sealed or otherwise. Have a look.
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