Based on skimming the light novel and the Wikia, this is what I could come up with about the summoning order of the Servants in the Fourth Holy Grail War. Fate/Zero doesn't give actual dates, but it does count down the hours until the start of the Holy Grail War, although the novel only gives them at the start of a chapter, so there's a certain amount of imprecision. We also don't know exactly when Assassin and Lancer are summoned, since those summonings happen offscreen, although we have some pretty clear indications of where their summonings fall relative to the other Servants.
Assassin is summoned first. This is mentioned in Act 1, Part 3 of the first novel. It happens offscreen, sometime before that chapter, so we don't have an exact time, but Kirei tells Tokiomi that according to Risei, whose ability to detect Servant summonings is infallible, no other Servants have been summoned.
Rider, Berserker, Saber, and Archer are all summoned in Act 1, Part 5, which takes place at -268:22:30. Their summonings are stated to be "almost simultaneous" and no exact times are given.
As far as I can tell, Lancer is summoned near this time as well, but we don't see his summoning onscreen. I give my reasons for thinking so below.
Caster is summoned in Act 2, Part 3, which takes place at -222:24:48, about two days after the summonings of Rider, Berserker, Saber, and Archer.
It isn't clear exactly where Lancer's summoning falls. We know it was after Assassin's, but we don't know how much time it took Kayneth to discover that Waver had stolen his artifact and prepare a different summoning. But after stealing the scrap of Alexander's mantle from Kayneth, Waver runs off, flies to Japan, and sets himself up in a house before summoning Rider. It probably took Kayneth around that much time to make his own preparations. Also, the first time we see Lancer is just after Saber and Irisviel arrive in Japan. Saber is summoned at the Einzbern Castle in Germany, and she and Irisviel arrive in Japan by plane in Act 3 Part 2 at -167:27:03. They meet and battle Lancer that very night, so Kayneth, who flew in from England, must have arrived around the same time or a little earlier. So my sense is that Lancer was summoned around the same time that Rider, Berserker, Saber, and Archer were summoned. It may have been a little earlier since Kayneth was already in Japan when Iri and Saber arrived, but I don't think it could have been later.
So, in the end, we arrive at this list:
- Lancer (?)
- Rider / Berserker / Saber / Archer ("almost simultaneously")
- Caster (two or so days later)