This is (almost) a copy of a calc I did in a content revision post and is put here by the request of Spinosaurus75DinosaurFan.
The time dilation formula is given by the expression: Δt'=γ Δt, where Δt' is the time that has passed as experienced from someone who is moving at a constant speed υ and Δt is the time as experienced from someone at rest with respect to the moving frame of reference. The factor γ is a factor of speed (the Lorentz factor) and is given by the expression γ = 1/(sqrt(1-υ^2/c^2)), where c is of course the speed of light.
8 minutes and 46 seconds is 526 seconds and is equal to Δt, since that was the time frame in which events would have transpired for normal people. This is very likely to be a low-ball, since the bomb would probably have exploded in less time than that, meaning the resulting gamma factor is even larger, wich means that the speed is a larger percentage of light speed.
Approximately 20 min (EDIT) is the episode's runtime during which the characters were in Flash-Time, which is 1200 seconds and is equal to Δt' (rest frame as seen by Barry). Substituting in the original formula, we have:
1200=526*1/(sqrt(1-υ^2/c^2)) --> (sqrt(1-υ^2/c^2))=526/1200=0.4383 --> 1-υ^2/c^2=0.1921 --> υ^2/c^2=0.8079 --> υ^2=(c^2)*0.8988 --> υ~0.9*c
So Barry had to be moving at approximately 90 % the speed of light, or at 269813212.2 meters/second.
Note: In case of a high-ball, one could say that the bomb would have exploded in 5 seconds, meaning the following:
1200=5*1/(sqrt(1-υ^2/c^2)) --> (sqrt(1-υ^2/c^2))=5/1200=0.004167 --> 1-υ^2/c^2=0.000017361 --> υ^2/c^2=0.999982638
--> υ^2=0.999982638*c^2 --> υ=0.999991319*c