I doubt it, else the limit would not be £500.
As mentioned before Lloyds are happy with anything sub £5k with no notice (though they advise you to call them first for anything above 3.5k just to be sure), and no extra checks are done until 9-10k is taken out at once but they do want a days notice [but say that often they can still do it same day depending on branch] (beyond this they have to by law carry out some money laundering checks I think).
Nationwide are not exactly small, there is no reason why they cannot have similar limits to Lloyds.
If it looks like a bank, smells like a bank, and does things of a bank ... it is for all intents and purposes a bank, even if it wants to be called a building society.