My understanding is that Monday night was jam session night at Birdland, and that these were regularly (?) broadcast, so it's quite possible that Blakey and Russell were caught on tape over several Monday nights. But for Fats, June 30 is a very late date, as he died from TB, a very debilitating disease, a week later. To make matters more complicated (for me) I feel that the first excellent trumpet solo on the "long" 52nd Street Theme (5:39) from the Miles Birdland Jam Session dated June 30 is in fact Fats, not Miles.
Essentially I find it hard to believe it is 1950 Miles when it sounds so much like great Fats .
However, my intuition is probably incorrect if the date of the session is actually June 30 ...