Corrected entry: After the big fight between Team Cap and Team Iron Man, Steve and Bucky fly to the bunker where the others winter soldiers are sleeping. Tony takes Rhodes to the doctor, then goes to the prison where Falcon and the others are caught, then flies to the bunker and he arrives almost at the same time as Steve and Bucky.
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Correction: It all depends on the speed at which Cap and Bucky could find the base plus any additional time that they might have needed to take to stop/refuel/sleep/etc. versus Tony/Iron Man's top speed when he departs for the base. (Especially as the Iron Man armor is faster than many military jets by this point in development.) It's clear that at least some time has elapsed by the time Tony leaves.
I was wondering this myself so I did some calculations of travel times. The big red flag is the travel from Germany to the Raft, which is in the middle of the Atlantic. Tony is shown making this flight in a helicopter, but a typical flight of this type would probably take about 4 hours. Now, remember that Sec. Ross needed time to arrest Cap's team, transport them to the Raft, process them, and they appear to have been in the cells for a while by the time Tony meets with them. So if it takes Tony 4 hours to fly to the Raft, and he does so after helping Ross arrest Cap's team, then takes Rhodey to the hospital, and likely gets treated for his own injuries while watching Rhodey get an MRI, It would be generous to say that Tony arriving at the Raft would be 7 to 8 hours after the airport battle at the earliest.
Vader47000
A plane trip from Germany to deep in Siberia would take 10-12 hours. While the Quinjet is much faster than a commercial flight, we see Cap is flying lower to the ground, probably to avoid whatever detection systems can be deployed to find the jet (assuming it has stealth capabilities to hide it from standard radar, but that Tony has advanced systems to find it). So while the quinjet is fast, a lower altitude would lengthen flight time. If Cap is keeping the plane under Mach 1 to avoid a detectable sonic boom, then he'd be going 600-700 mph, which would take 11-12 hours to fly to the Hydra base.
Vader47000
If the Iron Man armor can go Mach 5, it would take Tony about 2 hours to fly from the Raft to deep into Siberia. So if Tony leaves the Raft 8 or 9 hours after the airport battle, and it takes him 2-3 hours to fly to SIberia, then he is arriving in Siberia 10-12 hours after the airport battle. Thus, the idea that he arrives not long after Cap and Bucky is actually a plausible timeline.
Vader47000