Near as I can tell the siding at the NYCRR Hyde Park station was to the East of the mainline tracks and South of the still extant station building.
At about 1:15 in this video you can see the siding with some boxcars (ca 1920's ?)
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In that film there is what appears to be an "underpass/tunnel" under the mainline to a station platform on the west (along the Hudson River) side of the tracks. It seems that later the west side (railroad "east" towards NYC) platforms disappeared.
It seems that FDR simply got in an automobile at that station and took the usual roads up from the station and through the town of Hyde Park to his estate. From the current satellite maps it looks like maybe 4 or 5 miles away ? 10-20 minutes at 30 mph.
Maybe the railroad "spotted" a Pullman car on the siding for FDR to use, but I suspect they simply ran a special train (an engine on the south end and one or two cars on the north end) up the line to Hyde Park and then reversed direction back towards NYC (with an engine in front and one/two cars with the president behind the engine) ? It was a stub end siding, not much switching could take place there. They could put the President's car in the siding and use push poles to switch the engine to the other end of the car ? Or they could send two engines up there to handle the switching, but that seems like a bit much.
When FDR was Governor of NYS they may have spotted a Pullman Car on the siding and picked it up to go on to Albany, but that seems like a lot of extra work. They probably just sent a special train out of Mott Haven (the passenger car servicing yard) with a "fresh" car for the Governor and simply stopped on the mainline tracks at the Hyde Park Station to pick up FDR. Usually a siding that is meant to support Passenger Cars has at the very least some "House Power", an electric hookup to keep the lights on in a parked passenger car. Most of the bigger stations had tracks with live steam and potable water hookups to support a passenger cars overnight.
Would be interested to learn more.