I'm a US citizen, resident in the US. What taxes would I have to pay if I inherit money from a British relative in the UK?
As long as the relative in the UK is not a US citizen and isn't considered domiciled in the US, there will be no estate taxes imposed by the US. You can see a much lengthier explanation on this subject here.
The summary is that as a US citizen or resident, you can't circumvent estate taxes by moving yourself or your money abroad, but a relative who genuinely doesn't have US ties (other than inheritors located in the US) will be covered through applicable tax treaties (the UK does have an applicable tax treaty and applicable estate taxes as Ganesh points out).
From the UK side, the estate may be liable to "inheritance tax" depending on its size - but this will be paid by the estate itself before any payment is made to you, so if the estate makes a payment to you, the whole of it is yours to keep.
The tax thresholds are a bit complicated and due to become more so, but at a minimum any estate under £325,000 will be exempt. If your spouse died first without using their entire allowance, or for deaths after 6th April 2017 where a family home is being left to direct descendants, the allowance is higher. Anything above the threshold is subject to 40% tax, which will be paid be the person dealing with the estate before they can distribute anything else.