I have been awarded some RSUs which will not vest for 3 years. My employer is a company registered in the US (although I am a UK citizen and work for them at their UK branch) and therefore the RSUs are US. When completing the US form for US tax exemption it asks for a Tax Identification Number. From googling, it suggests that we don't have one in the UK and the closest thing we have us a UTR (but you only get one if you register for self assessment). As I don't do self assessment, I'm wondering whether I'd be able to put my NI number down?

Such problems happen just because you have an RSU, and in the US, everything is strict with this. Apparently, you need to find out your company's EIN first so that you can convert your RSUs into real securities in the future. These are the rules of the USA, and you have to put up with them. I would advise you to get an EIN first and only deal with the tax authorities. Moreover, it has become much easier to get an EIN because of services like this -- https://e-delaware.com/ein that can quickly make you an EIN without problems. However, you should have some information about your company when you receive an EIN.