UK Citizen with US based income question...

If a UK citizen received a 1099 for income from the US, would they be liable to file both US and UK taxes?
They are not and have never been resident aliens or green card holders nor have they ever lived in the US.
They live in the UK currently.
Any help much appreciated...
Thanks

It depends on how much income it is, what kind of income it is, and how much tax was withheld.

Even if you don't have to file, you may want to file because you may be due a refund.

Interest - zero tax due under domestic law.
Dividends - usually 15% under the US/UK treaty.  There are many exceptions that increase and decrease the rate.
Rents - graduated rates, with expenses allowable.
Capital Gains - zero tax due under domestic law, if you are British domiciled.  There are a few exceptions.

If you post the amount of income and what kind it is, that would be helpful.

It was misc. income, and was $21,000 and no tax was withheld- I believe it would be classed as self-employed income?

You are correct; as misc income, it's self-employment. 

Under the US/UK treaty, if your business does not have a "permanent establishment" in the USA, you can pay the tax in this country and owe no tax to the US. 

If you have a "permanent establishment" in the USA, you owe tax in the USA (but no self-employment tax), and you also owe tax in the UK.  The US tax paid will be a tax credit in the UK, so you won't owe more than the UK rates.

In either case, the amount of income requires a US tax filing and specialist knowledge to determine whether you meet the permanent establishment rules or not and application of the treaty.  I recommend seeking a professional to prepare your tax returns for the year in question. 

The cost of hiring a professional to prepare these business-related returns would be tax-deductable.  You can contact me through my website, www.BritishAmericanTax.com, or look for other professionals on the web.