I do not think there is anything to confirm the month of the 14th
The movie was mostly filmed in the period July through early September in the northern hemisphere. Locations like the wind-farm were beginning to look somewhat less warm. So it might be intended to look like late summer / early Autumn.
I don't believe there is any specific connection between the "14th" and "June 29th", primarily because the party at the Embassy is being used purely as misdirection or alibi.
It is a party that Kat and Sator attended, and it is being used as an excuse for why Kat knows the Protagonist
It won’t be long. In the meantime,
I’m a former First Secretary from
the American Embassy in Riyadh, we
met at a party last June –
We were at a party in Riyadh, but I
don’t think it was June –
June 29th. 7 for 7:30, salmon on
the printed menu swapped for sea
bass on the night. Sator left
early, that’s when we met. Don’t
offer any of this, it’s there if he
asks. I came into Shipley’s in
London, you ran into me here and
want to show me the yacht.
I don't think this is intended to imply the June date is somehow important to the plot, or that the Protagonist was actually there. The details of the party he is using here are presumably from research made by the Tenet organization.
The use of "last June" could even refer to June of the previous year.
In any case, I don't think it's particularly important.