There's an additional problem when selecting external pipe insulation because it has to be UV resistant of course.

The range of sizes available for installers to choose from is therefore reduced,
and especially so if they wish to use larger bore insulation around valve bodies.

Pipe Lagging offer a selection of 'solar' pipe insulation online, but it's expensive once you opt for sizes outside the common pipe diameters.

Another option is to swathe UV-resistant tape around the odd-sized sections and bends.
But my own experience with this approach is that the adhesive on the tape isn't good enough to hold firm around tight corners.
It needs supplementing with contact-adhesive at the edges.

There are also anti-UV paints which can be applied on top of existing insulation...
... provided it hasn't already been degraded of course!

The relevant sections are 4.24 to 4.26 of approved document L: https://assets.publishing.service.gov.uk/government/uploads/system/uploads/attachment_data/file/1133079/Approved_Document_L__Conservation_of_fuel_and_power__Volume_1_Dwellings__2021_edition_incorporating_2023_amendments.pdf

It's worth noting that the insulation thicknesses described in that document are for very high quality insulation (i.e. very low thermal conductivity and low emissivity). The building regs defer to BS 5422 for lower performance insulation, and that can have a huge influence. See the attached two tables for low and high emissivity insulation, cribbed from a verv old copy of BS 5422.

 If we take 22mm pipe as a reference. The really high quality insulation would require only 8mm thickness, but if it was high emissivity then that would increase to 11mm and if it then had high thermal conductivity typical of the sort of neoprene like insulation (~0.04 W/mK) then you'd need 23mm insulation to meet the regulations. That's a big jump from 8mm!

As an aside, the building regs are a great source in some areas like this. IMHO they're easy to read and a model of clarity.

To seal where pipes enter the steel shed, I'd be using offcuts of EPDM rubber roofing material.
It's excellent for shaping around joins, and easy to cut.

You can glue it directly onto existing vertical surfaces.
So it's also a good choice to seal the top of your boxing.

There isn't enough clearance between the pipes and wall to insulate it much better.

My sons arm was recently injured quite badly, but as soon as he is fit I will lob him up in the loft to check the pipe insulation in there. I wouldn't be at all surprised if it is just the rockwool!

To calculate this, first ask them how long it will take them to remove what's there and do it again properly.
Then use £45/hour per person and add in the cost of buying the insulation yourself.
That should provide you with a good starting point.

They'll be super busy and wanting to get on with their next job rather than fix yours.
So the option of repaying you in cash will be tempting for them.

One possibility for you to consider is to make a rectangular box around those external pipes and seal them in...
... at least for the main vertical run.

There are PVC parts used by window-fitters and firms who clad facia boards, which could be fashioned into suitable boxing.

However, I have taken some time to check what you guys are saying and you're entirely correct.

See pics!

My first avenue was simply to get it all corrected, however I see now I will have to find the time to do most of it myself if I want it done well.

If I posted pictures of the horrific work they did in my house I think you guys would turn up with pitch forks outside their office!

It must be draft-proof,
which may require neoprene rubber gaskets to be cut and glued around the pipe-exit holes.

I can provide further construction details if required.

The adhesive will be a 'contact' type such as this from Screwfix

Pipe insulation is usually best undertaken as a DIY task...
... preferably by someone with OCD!

Although I can't see how they arrange themselves to pass through a 100-125mm outer sleeve, the pipe clips have already compromised the wall thickness before the flow even gets there.

I also wonder if the installer has used Split-Insulation.
If so, then the split will be gaping open on the side which faces into the wall.

The approved method is to use adhesive to join the splits,
and for all bevel-cut ends around pipe elbows.

This then gets an outer layer of 'tape' to further cover the joins.

Armaflex and Kaiflex both manufacture UV-resistant external insulation.
It's usually sold as "solar insulation" because it gets used on roof-mounted solar-thermal collectors.

Like your house, ours is slow to hear and cool, having all concrete floors. Our kitchen/living room is 15 metres long of South facing patio doors, so we get massive solar warming, and I have to turn down the AAA thermostat from say 20C to 18C as soon as the sun comes in - turning it down by less does not quickly stop the heating. Likewise, it's best to turn it back up before the sun goes down to reheat the floors. I hope these experiences are of some help.

Also regarding the loft insulation, I don't know, I haven't been up there since they did it as I have a severe back problem and emphysema which is why I qualified for the grant. My guess is they just buried everything.

I did take the precaution of running most the electrical cables along raised joists when I rewired this house. It is very old, still had black and red twin and earth cables most of which had been butchered at some point.

Even 10 degs of thermal loss wouldn't account for this though, on a good day its 45 degs, on a cold day its 25 at best. I am in a rural seafront location, open fields over 300 metres between me and sea. I suspect its simply to cold in the mornings for the WH to overcome it in the heating cycle which I think is 1.5 hours.