Doing literally that is a nightmare, as an electrician can't do that without either working live (dangerous and therefore illegal per HSAWA/EAWA) or removing the main fuse, cutting the security seal (which only the DNO or those with delegated authority like meter operators are allowed to do).
Conversely, DNO/meter operators don't permit their staff to do work on customer-owned equipment like consumer units (because they would be assuming liability for stuff that they don't have any control over). So the legal way to do it is to arrange for DNO to remove the fuse, electrician to do the work, DNO to put the fuse back - a logistical nightmare.
The more practical option is to get an
isolation switch fitted between the meter and consumer unit: now you have a clear split between your equipment and the supplier's, you can turn off the switch and work on everything you own on your side of it, and they can work on their side of it.
You can usually get this fitted by your supplier (who will probably contract their meter operator to do the job), at prices varying from free to substantial. Alternatively, having one in your back pocket when having other work done (like when a smartmeter is being fitted) and asking the fitter to wire it while they are on the job is usually a cheaper option.
Note that the above applies both to the situation where you need your tails upgrading for a 100A fuse and also for the case where you are splitting the tails to have a separate consumer unit for the EV charging and leave your existing consumer unit alone - the latter again needs work on the tails which can't be done safely without shutting off the power.
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As to the question of whether your tails need upgrading, that depends what main fuse sizes you are talking about. 16mm² tails (most common size) are good up to 80A (if exposed on a wall), so the upgrade from 60A to 80A fuse doesn't typically need the tails upgrading; however going to 100A main fuse does.
UKPN are keen on 100A fuses, so this is most likely to be the situation if you live in a UKPN area. Other DNOs are less keen on 100A fuses, and some won't go above 80A at all (insist that you upgrade to three-phase if you want more than 80A). So experience in one area doesn't necessarily apply to others.
This is mainly a problem if you want more than one chargepoint and/or electric showers. For a single chargepoint, most ordinary households will fit within 80A, and many would be OK with 60A. For cases where it is a problem, chargepoints that can do power management with a sensor in the incoming supply are the answer - Zappi, eVolt, and EO all offer this (and probably more that I haven't noticed).
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