Well, for all we know, Illya summoned Berserker two months before the actual war, which will fall on the month of December or last days of November, therefore she is the earliest summoner of a Servant in a war, and a world record of early summoners for the Holy Grail War.
It's possible that Sakura summoned her Servant either 3 - 4 weeks or way more before the actual war.
Emphasizing this word in this sentence for great detail:
Assuming Bazett summoned her Servant before Sakura, the order will be this:
Illya, unknown Magus, Bazett, Sakura, Caster, Rin, Shirou.