Looking at the Rubik's cube records page, it seems there's no speed record for completing a tactile 3x3 Rubik's cube. There's a record for doing a standard cube blindfolded, but that's a record for sighted people and it includes the time it takes to memorize the cube's layout. It's surprising there's no tactile record. So I'm declaring myself the current world record holder with a time of 3 minutes 6.45 seconds. If you want to snatch my record you'll need a touch cube an iPhone stopwatch, and the honesty to properly jumble the cube, self-police your record attempt, and keep your eyes closed if you have any useful sight.
The rules are you can position the cube with the smooth/white centre piece at the top of the cube, and have a quick feel around to assess the cube and see what your first move will be. Then you tap start on the iPhone's stopwatch, complete the cube, then tap stop. 3:06:45 is your target.