Can an American get into the UK if they had a previous DUI?

Yes. The UK will deny entry to travellers who have been convicted of an offense in which if committed in the UK would be punishable by imprisonment of 12 months or more.

You can even get entry to the UK with a felony conviction as long as the conviction is 'spent' which is generally considered to be when a conditional discharge from probation occurs or one year whichever is longer.

UK also makes exceptions for 'exceptional circumstances'

If you are concerned, you should contact your closest UK visa issuing office http://www.ukvisas.gov.uk/missions