Does 1957's 12 Angry Men consist of only one cut?

Assuming you're referring to the 1957 B&W version, the simplest answer is that the film was indeed unique (for the time) in consisting of some extremely long mobile sequences with a mounted camera. The initial scene with the jury introduction, for example is nearly 10 minutes long and involves no less than 30-40 different conversations in the foreground and probably ten times as many in the background.

That being said, there are cuts throughout the film, for example the one below at timecode 00:10:45.

If you're referring to "takes", several critics and commentators have noted that the film is famed for having required just 365 separate takes (and 2 weeks of solid rehearsals) in order to get the film in the can. When compared to some filmmakers (Kubrick famously took 148 takes to get a single scene right in The Shining), this can be seen to be nothing short of miraculous.

Richard has done a great job of explaining that it is more than one cut but I'd like to add why.

The fact is that, even today, it's pretty much impossible to make a feature-length film in one cut... even with digital recording.

In the 50s, it was even more limited. All films were shot on actual film and filmmakers had to work around the limited length of the reel - either 500 or 1000 feet - which, at 24 frames per second limits directors to 11 minutes of shooting time per shot.

Modern digital cameras are limited in their shot length by the size of their digital media and the recording quality.

As an example, here's the stats for the Arri Alexa.

To paraphrase, with a 64 GB card, you'll get between 24 and 210 minutes of recording time depending on the codec you use. Now, 210 minutes is a really long time but the quality at this level would be very low. Some cameras may also have issues with overheating when used on very long shots and, if running on batteries, the batteries would certainly not last 210 minutes.

So, while possible (and actually accomplished in a 96 minute long Russian historical drama called Russian Ark), it would take some extreme measures to actually do it. For example, to limit the number of restarts, the director of Russian Ark recorded video only to avoid sound issues and recorded directly onto a large external hard drive that was able to hold up to 100 minutes of uncompressed high-def footage and had to be carried around with the camera.

What I love about this movie is that though the set was a room Lumet and Kaufman, his cinematographer, did their best to amplify 12 Angry Men tone so that at the end the feeling of tension appears. Also, Lumet and Carl Lerner, his editor, used editing techniques to increase this tension. At the beginning there are quite long takes but gradually as the conversations start heating up, cuts start coming faster. All in all there were 360 shoots which were carefully thought through from the very beginning till the end.