And the same museum at Bletchley Park (there are two) is building a functioning replica of the EDSAC, one of the first programmable, fully electronic computers:

http://www.tnmoc.org/special-projects/edsac

(Whether EDSAC was "the first" depends on how you define "computer", and what you think it means for a machine to be in operation, among other things, but it's one of several plausible claimants. The Z3, cited elsewhere here, was a programmable computer, but not fully electronic -- there was a lot of relay circuitry, which kept the clock rate down to about 10 Hz.)

From the title, I was honestly expecting the article to be about this project instead.