Barty Crouch Sr. was being ruthless, and Sirius didn’t help his case.
When Sirius is telling the trio about his experiences with Crouch, he explains that plenty of people were sent to Azkaban without trial when the war was in full swing:
“Oh I know Crouch all right,” he said quietly. “He was the one who gave the order for me to be sent to Azkaban — without a trial.” […]
And I wasn’t the only one who was handed straight to the dementors without trial. Crouch fought violence with violence, and authorized the use of the Unforgivable Curses against suspects.
— Goblet of Fire, chapter 27 (Padfoot Returns)
If this was common practice, it’s less surprising that nobody advocated a trial (or was unsuccessful in doing so); such an appeal is likely to be unsuccessful. And remember, Dumbledore has no reason to believe Sirius is innocent; indeed, he had suspected a spy in the Order for some time.
If this order came from Crouch at the Ministry, Sirius’s allegiance to the Potters or the Order of the Phoenix counts for nothing; that will have no sway on Crouch. If anything, the fact that this was a betrayal may count against him – spies are always looked upon badly.
(In some wars, captured enemy spies have been subject to summary execution – execution without a trial. This seems to be the magical equivalent of that.)
And here’s how Black reacted to Peter’s “murder”:
“And Black standing there laughing, with what was left of Pettigrew in front of him… a heap of bloodstained robes and a few — a few fragments —”
— Prisoner of Azkaban, chapter 10 (The Marauder’s Map)
Combine with multiple eyewitnesses, and a suspicion of a spy among the ranks – Sirius fits rather neatly. If (as is suggested), he did nothing to profess innocence, it’s very quick and easy to send him to Azkaban, and save yourself the hassle of a show trial.