If they had a large enough negative interest rate for it to leech through to Joe Public, I wonder how long it would take Barclays to pay off my overdraft for me?

Never going to happen.

It'll never happen because, for instance, contracts like the mortgage deal I have with Nationwide (who I'd recommend as a building society) are written to ensure that it'll never happen, my base rate mortgage is set at 2% above the BoE base rate with an absolute minimum of 2.5% (which is where its at now, I think), so no matter how low the BoE go now my mortgage rate won't reduce any further.

New mortgages with Nationwide are even tighter with a floor of around 3.5% I believe.

Not only that the proposal is to split rates and not reduce Base Rate but only the rate the BoE give / take from banks.