Still looking, but here is a partial for now:
TL;DR: October 9th appears to be the date!
A Canadian took the first photograph of Sputnik from North America on October 9th.
Also on October 9th, Australian newspaper The Age, publishes a photo from the night before, claiming it is the satellite. If so, it might well count as the first from that side of the world.
An observatory in Canada lays claim to the first North American photographs of the Sputnik I satellite:
In October, 1957, the Newbrook Observatory bore witness to one of the seminal moments of the twentieth century. On October 4, 1957, the Union of Soviet Socialist Republics (U.S.S.R.) astounded the world by announcing the successful launch of Sputnik I, the first artificial satellite to be sent into space. Less than a week later, Art Griffen, resident scientist at the Newbrook Observatory, took the first North American photograph of Sputnik, confirming the Russians' claim.
I note it says 'North American' which leaves space for somewhere else to claim the first ever, anywhere, photograph, perhaps.
Shortly after the launch, Griffin was asked by an American official to mark the satellite as it passed over the observatory, perhaps to confirm that the Soviets had, in fact, beat the Americans to the punch. In the early morning hours of October 9, 1957, Griffin caught sight of a thin beam of light that he thought could represent the flight of Sputnik I through the upper atmosphere.
The Age - Oct 9, 1957
Bright Spectacle in Clear Sky
THE Soviet earth-satellite and part of the rocket
which placed it in orbit provided another brilliant
spectacle in the sky over Melbourne last night.
Thousands of people at widely scattered places from
Tasmania to Brisbane followed the path of the main object across a clear sky for about four minutes. Leading physicists Dr. F. Jacks and Dr. V. D. Hopper each saw the second object and expect the Russian satellites will appear at the same time tonight--about 7.38.
Switchboards of newspapers and radio stations were flooded with calls many of them from people who saw a second, dimmer object following the course of the other on a slightly different course a few minutes later. The federal secretary of the Wireless Institute of Australia (Mr. L. D. Bowie) received reports of sightings magnificent radio amateurs in four eastern states.
Reports came from widely separated places including Brisbane, Newcastle, Wagga, Coolamon, Wyong, Yallourn, Horsham, Ballarat, Bright and Ulverstone in Tasmania. Mr. Bowle said the radio signals from the satellite seemed to have broken down about 4.54 p.m., when they stopped abruptly. Radio amateurs believed that if the satellite was still transmitting, it had changed to a different frequency.
Dr. Jacka, who is Chief Physicist of the Department of External Affairs
Antarctic Division, said he saw the first object rise silently east of south at 7.38 and followed it until it disappeared in the north. The second object rose in about the same position, but was much fainter and proceeded more to the west. Dr. Jacka said Russian statements about the rocket being larger than the satellite seemed to indicate that the first, brighter object was the rocket, and the second object the satellite. The difference in apparent orbit between the two objects was due to the which Earth's rotation, would cause an appreciable difference in only a few minutes.
Dr. Hopper, Reader in Physics Physics at Melbourne said he followed both objects with a theodolite and took readings of their azimuth and elevation.
The second object appeared about four minutes after the first, and was slightly reddish in appearance. It disappeared when roughly overhead.
The time lag between the two objects tonight was likely to be slightly greater.
This is the first official photograph of the Soviet Satellite Rocket (note: not the satellite), using the IGY satellite tracking camera at South Pasadena, California at 5:06 a.m. PST, October 17, 1957 by the Smithsonian Astrophysical Observatory personnel. The rocket appears in the lower part of the photograph.
The rocket stages were photographed more than the satellite:
Although not in orbit by then, LIFE photographer Robert W. Kelley captured this, with a possible incorrect date (February), from Montreal, Canada. LIFE and other magazines stationed photographers around the world in anticipation of catching a glimpse of the satellite on film.
The following is also attributed to him:
Video footage below claims to be of the actual satellite:
NBC News Archives Clips
Date created: 13 October, 1957
Again, rocket stage photograph:
Schenectady Gazette - Oct 15th 1957
7th October, Moonwatch still looking for satellite: