I've run a quick search on the Companies House website and there is definitely no company with that name registered in the UK. It might still be the 'trading name' of a legitimate company but, if so, I'd expect that there would be a proper website, under that trading name, for the firm. All I can find is a site at
http://www.natbbuk.com/ which seems to have some files on the server but no home page.
NatWest Asset Management Holdings Ltd is definitely a legitimate firm (and a member of the NatWest group). If you phone them on 020 7726 1000, I'm sure that they will know if the bank you mention is also part of the NatWest group.
Chris