I think there are two points to make. Firstly your husband's employer cannot insist that you keep lots of money in that offshore NatWest account. They might be able to insist that he has that account, and that his pay goes into it, but whatever happens to the money thereafter is none of their concern.
But, knowing something of tax arrangements for expats in the UK, if indeed that is what your husband is, your husband should get tax advice before remitting any of the money into the UK. That could be very expensive in terms of tax.
Secondly, NatWest is owned by Royal Bank of Scotland. It is a big secure bank. It has guaranteed all deposits made into its CI subsidiary. RBS has benefitted from the UK Government announcement today. Or put it another way, if RBS is in real trouble the whole financial system would be in collapse.
But nothing to stop money being moved around, in a tax-effective way. Maybe put money into an Irish bank?