Will his income be liable to US as well as UK taxes?
Yes, BUT ...... double taxation relief will be due. That is, the US tax will get reduced by the amount of the UK tax on that income, or the US tax on that income, if lower.
For example, if the UK tax is the equivalent of $5000, and the US tax on the income (prior to double taxation relief) is $6000, then in fact the US tax on the income will be $6000 - $5000 = $1000.
Second example, if the UK tax is the equivalent of $15000, and the US tax on the income (prior to double taxation relief) is $12000, then in fact the US tax on the income will be $12000 - $12000 (capped) = $0!
Those are the general principles that will operate.