When I got a call from Santander, telling me that there had been suspicious activity on my account, I was extremely wary of telling them anything (and nearly hung up). However I found out that a total of £800 had been withdrawn from my account, at cash machines nowhere close to me, without my knowledge. (Fortunately I got it back without too much hassle).
So you might find that the call is genuine (particularly in view of their persistence) and that you need to act to protect (or recover) your money.
Of course the apparent flaw in my argument is that you don't actually have a NatWest account. However if you've got an account with RBS, Ulster Bank, Drummonds Bank or (if you're like the Queen!) Coutts, then you may well still need to act. They're all part of the same group as NatWest and may well coordinate their security arrangements through NatWest.