No, rather, the UK PAYE (tax withholding) system is more sophisticated than the US one. Most people have exactly the right amount withheld from their UK pay. Thus, there is no need to file. If you are self-employed, obviously no one is withholding your taxes, so you'd have to pay. And in your first year in the UK, the PAYE is sometimes overwithheld, and you might need to file to get a refund.
As long as your income from investments is offshore (the USA is offshore to the UK), and you do not remit the money to the UK, you won't have any UK taxes on your investment income. Thus, with only your wages being taxed, and the PAYE system being so amazingly accurate, you won't need to file.
Learn more about remittances here:
http://www.hmrc.gov.uk/pdfs/ir20.htm#investmentLearn more about your nondomicile and residency status here:
http://www.hmrc.gov.uk/pdfs/ir20.htm