Maintaining a large paper archive is almost free in the grand scheme of things. The UK (and pretty much every single other Western nation) has an absolute ton of abandoned bunkers, silos and mines that were intended for Cold War disaster management. Most of them were built to very high standards in locations that won't flood or degrade significantly even if left cold and dark.
Using them for archive storage is trivial.
This seems like a very poor decision. We're seeing vulnerability after vulnerability pour in this year in just about every single OS out there and yet they think something like this is a good idea?
The UK government appears to be trying to shut the country down without anybody noticing until it's too late; it's very weird to watch.
Yes, it isn't zero-effort, you can't just chuck your papers down a mineshaft and come back for them 50 years later, but it isn't in any way an insurmountable problem.
Fire, one of the biggest concerns in archival practice, is also much harder to manage in underground facilities. Document archive facilities are designed as distinct fire cells to mitigate the risk, and incorporating these features into bunkers would likely require rebuild from scratch. Refrigerant fire suppression systems are difficult to install because of inadequate evacuation routes; Cold War era underground facilities rarely have more than one staircase and one ladder (and the ladder shaft might be full of sand!).
The disused military bunkers of the United States are littered with failed document storage companies that have been unable to sufficiently retrofit them for this use. Seriously, I would guesstimate as many as one in six former ICBM silos was, at some point, owned by one of these operations. More than one per field usually. Almost none have succeeded, and the exceptions have the appearance that they are more hobby projects than viable services. In most cases they didn't even finish stripping the old equipment, which is a very costly process since it has to be cut up underground to hoist out.
Perhaps records only need to last lifetime + 70. That should make storage concerns a lot easier.
Here in Germany, a lot of our old bunkers are effectively gone, unusable to a point where you'd need to entirely rebuild them. And that's after just 30 years... bunkers also need maintenance, because otherwise rot and mold will set in.
A4 is 1/16th of a square metre, so on 80gsm paper with a density of 1.2x water, 1MW (mega will) is about 25m³. An ISO standard TEU shipping container has a volume of about 30m³.
If you spaced everything out 10x you’d still only need 650 containers for all of the UK. That’s about 3% of the volume of a regular container ship, or roughly the number of containers you might see in a Hollywood standard bad guy dockyard confrontation movie scene.