When building works are done on a house in the UK (specifically England, if it makes a difference), is the contractor who erects scaffolding required to have public liability insurance? If the scaffolding fails, and they don't, would the home owner be liable for any injury/damages?
No one is required to have Public Liability Insurance
If a contractor (or householder) wants PL insurance they are free to take it out but there is no legal requirement to do so.
There may be contractural requirements. For example, a mortgagee may require a mortgagor to have property insurance and PL usually comes bundled with this (but check your policy because it may not be so). Or a householder principal may require their contractor to have PL insurance but a wise principal also requires evidence of cover.
Whether a person has PL insurance or not, they can be liable to the public for damage. Insurance does not remove liability, an insurer just indemnifies their insured from the costs of defending or settling a claim.
In general, the head of claim for liability is typically negligence. Whether a person is negligent depends on if they had a duty of care and failed in that duty among other things (see Is there liability for pure accidents?).
The contractor could be liable. The householder could be liable. Both could be liable. Neither could be liable. In normal circumstances, a plaintiff will sue everyone they can think of. At that point, it would be nice to have insurance so you can say to your insurer: "You deal with it."