First I will point out that the .24 HT to FAT conversion applies to a "real" hand time. This means a trained timer who starts the watch with the bang of the gun, then stops it by carefully watching for the athlete to cross the finish line and standing even with the finish. The timer needs to be near enough to the start that the time of the sound traveling is not significant (within about 20 meters, so under a tenth).
Im assuming that your friend didnt start it based on when a thirt party fired a gun, but either said "bang" himself, in which case he started the watch earlier than a normal HT, or started it when he saw you begin to run, in which case he started it later than a normal HT.
My point is, your time is likely either faster or slower than reported by about .20 either direction (assuming average reaction time, since elite sprinters have reaction times about .17 on average).
So that puts you at either 7.27 or 7.67 for an actual HT. Therefore, about 7.5 or 7.9 for a FAT time.
7.9 for the 60m is about 12.3. 7.9 for 55m is about 12.9
Of course, 60 to 100m conversions are tricky, epecially outside of the very elite. Elite sprinters tend to all be elite in their reaction time, acceleration, too speed, and speed endurance. This means that the conversions are fairly reliable, since for example, somebody running a 6.5 60m most likely is going to continue along the final 40m of the 100m at an elite range of 12m/s or faster. Very small differences will be found between two different 6.5 60m runners in that final 40m. Those small difference separate the olympic finalists from the heat fillers.
In the case of a runner far outside of elite, huge differences may be found. For example, a 320 pound NFL lineman might run a 7.9s 60m by blasting the reaction time, having a pretty solid first 20 or 30m, but then being at max speed at the 30 and beginning to fade already at the 60m mark.
An elite distance runner might run the exact same time in the 60m by having a terrible reaction time, awful acceleration for the first 30m, but then 30 to 60 be rapidly closing the gap to that giant lineman in the race.
They have the same 60m time, so their 100m time prediction will be the same from the conversion only. Yet that distance runner going through 60m in 7.9 may run near 12 flat while the lineman comes chugging through in 13.8
Realistically, a "typical" distance runner runner 7.9 I would predict about 12.7, while a distance runner known for very good top end speed and efficiency might be in the low 12s