I am not sure that you are correct.
From what I have (since) read elsewhere, the OP is a involuntary bailee (Torts ((Interference with Goods)) Act 1977). They have to give written notification that they plan to dispose of the items and that they have 28 days to collect the property. The homeowner is then obliged to hold on to the property for 3 months. If they sell the items they can deduct any costs related to the sale. The owner of the property has up to 6 years to ask for those funds. Anyone buying the property becomes the legal owner.
I am not a lawyer but the fact that the OP believes that the property belongs to a third party seems to muddy the waters.
The op doesn't know who is the owner of the scaffolding.
If it was me I would use due diligence and call all scaffolding companies in the area.
If I had no luck finding the rightful owner I would give the cowboy builder 2 weeks notice to arrange removal (set a deadline).
The scaffolding company has a contract with the cowboy builder, not the landlord.
If the scaffolding disappeared, they would sue the cowboy builder, not the op because they have no contract with them.
At best they could list the op as co-defendant, but it would immediately fail when the op produces evidence that they tried their best to return the scaffolding to the rightful owner.
The cowboy builder could sue the op for disposing of the scaffolding and subject them to a breach of contract.
This again would fail immediately because the op has given them notice.
County court, where such things are heard, do not deal with nonsense.
Blackmailing a customer would surely not seen as a good move from any judge.
Given that it ever reaches court, which it's near impossible given the circumstances.
The op just needs to be firm.
Give a ring to local companies.
If no luck, give the cowboy builder 2 weeks notice, then put the scaffolding up on the net for free and send a link to the cowboy builder.
They'll understand that you're not kidding.