Hi Fasalldes

There are a few things to consider here.  Simple answer is that thermawrap won't be enough.

Now the complicated answer.  Basically you want the temperature in the shed to be comfortable while you're in the hot tub.  Something like 21 degrees is fine if you're sitting in an armchair, but you'll probably want it a bit warmer if you're in the tub, and you particularly don't want drafts because wind-chill is much worse on wet skin.  So let's say the shed needs to stay at 25 degrees Celsius and drafts should be minimised.

The more insulation you have the less you have to heat the shed, so lots of insulation is good, and that also cuts down convection drafts too, except insulation costs quite a bit of money and it takes up space too - not insignificant in a shed.  For a rough idea, a 10 x 12 shed insulated to U=0.2 floor, walls and ceiling might lose heat at the rate of 450W when it's -5 degrees outside and 25 degrees inside, which on a green tariff will cost about 6.5p/hour.  I think thermwrap has a U-value of 8, so the shed would lose heat at something like 17kW and it would take an industrial furnace to heat it.  It's a rough calculation but it gives you some kind of idea - I assume walls, floor and ceiling conduct at the same rate, which isn't true, and that a third of the heat gets out some otherhow.

So if you're free to choose how much insulation to use - I'm assuming building regs don't apply - then it's helpful to know how much it costs to insulate against how much it costs to heat.  For example, insulating the shed with 100mm thick Kingspan TF70 sheets would cost at least £550  (current price on ebay) and would give a U-value of .23 which makes the heating cost something like 7.5p/hour on the coldest day of the year, but much less on average.  Say you heat the shed for 10 hours a week all year with an average outside temperature of 10 degrees, you'd expect to pay less than £20 for the year.  If you plan on using the shed for a good few years then £550 isn't such a bad investment, especially not if heating prices go up, which they're doing seriously at the moment.

I've ignored a lot of details here, not least the heating contribution of the hot tub, but it should give you an idea.

Anyway, what you also need to consider is that a hot tub will create an increadibly humid atmosphere, and that's a big challenge.  Good insulation is essential to stop condensation forming on the walls and creating damp and mould, but you need to be really careful about cold-bridging too.  You also have to incorporate a vapour-check barrier behind the internal cladding and make it totally leak proof otherwise damp air will get into the structure and condense and this will rot the building.  Sealing this kind of humidity up against drafts is bad news because it will get stale very quickly - you'll probably need something like a passive stack vent, and maybe even a heat-exchanger to heat the incoming air and recover some of the outgoing heat.

Do check the figures, I might have got it wrong, but I hope that's helpful.