I suspect you have a complaint against the solicitor for the way they've handled your transaction, but not a financial claim, if you can't prove that they were aware of the remortgage.
This is a pretty unusual situation, it's almost unheard of to transfer your mortgage so close to selling, so it's likely that both solicitors were following the normal procedures. You'd need to prove that your solicitor was aware that you'd remortgaged, and therefore the standard procedure wasn't enough.
How many people are in your chain? Is it just you and your buyer?
Solicitor A can ask the Land Registry to expedite the case as it's holding up a transaction - it'll then be processed as quickly as possible, but I doubt that will be before Tuesday. It's unlikely that Nationwide would allow the charge to be cancelled, as their charge would then be unsecured, even if you were imminently planning to pay it back...
An undertaking could be an option, if your buyer and their solicitor are happy with that, but it's likely to be a no-go if your buyer is getting a mortgage, as that usually requires the bank to be the first charge.
Realistically I'd have said your best option is probably to get Solicitor A to expedite their LR request; and prepare the rest of the chain for delayed completion, but you could incur costs for that - so the size of your chain is pretty important there.