If an American were to marry a Brit what would happen to their citizenship if they got divorced?

Nothing, you don't automatically become a citizen of the other person's country because you marry them, there's a whole of legal hoops to jump through first. It can takes years to gain citizenship of the UK. For a spouse to gain citizenship in the UK, they must be legally resident in the UK for a minimum of 3 years then apply Indefinite Leave to Remain (ILR), once ILR is granted you must wait a year before you can apply for citizenship by naturalisation. In theory once ILR has been granted you could divorce, you could even separate during the period leading up to applying to ILR but you would need to satisfy immigration authorities that the marriage was genuine - if the split is acrimonious & the British citizen decides not play ball there won't be an issue around citizenship. Generally speaking once citizenship is granted it isn't usually revoked, unless it can be proved beyond a shadow of a doubt that the marriage exists for the sole purpose of defrauding immigration & gaining residency for one or other of the spouses, in which case you'd both be in sh*t, presumably there are crimes which allow the authorities to strip a person of their citizenship & deport them back to their home country.