I also saw this Home Again with the butcher block brick floor. I even taped the segment.
I noticed that is project was from 1991. I would love to know how well this kitchen floor held up. (The project was the Carriage House in Chicago.)
The grout was described as 2 parts sawdust to 1 part "fast drying oil based sealer". This was sqeegeed off and lightly steel wooled while still wet. Then the floor was sealed again 24 hrs later.

I have seen the same cobblewood or endgrain block technique used in my area (South Texas) using mesquite, a native hardwood. I'm interested in laying such a floor in a small room in my house but don't know what method should be used to attach the floor to the underlayment. Adhesive? Nails? Screws?

Any suggestions will be appreciated.

The Home Again episode I saw from 1991 used a non-water based "rubber mastik". I believe they commented that a water based adhesive would warp the woodblock tiles. It was troweled as you would tile (I think it was with a 1/4 inch square tooth).
I still would love to know how durable this application is.