This is an awkward one. I assume this was not the first payment being made and it had been running smoothly for some time. Unlike a direct debit, the onus is on Barclays to make the payment correctly, using the correct NatWest sort code and your daughter's account number.
I know it's time consuming, but you need firstly to ensure Barclays made the payment correctly and on time. Tell them because of the delay that incurred, charges were applied. Ask for the proof that this was paid and to put it in writing to you.
Provided Barclays can do this, then the blame will lay with NatWest. But if they can't, then you need to press Barclays for repayment of any charges incurred by your daughter, but show them proof of the charges, so they can't wriggle out of paying.
Now if Barclays can show you there was no error on their part, then you will need to put pressure on NatWest. Ask them to prove to you on what day they received the standing order payment. Was it received late, or did they receive it on time, but initially pay it to an incorrect account perhaps.
In reality, standing orders should be safer than direct debits, because the instruction is made to the bank whereas with direct debits, you're giving permission to a company to help themselves to your money.
Whatever you do don't stand by while each bank blames the other, one of them is in the wrong, and when you dicover which one then press at the very least for a refund of charges plus you could try for a payment for all the hassle involved