From this Wikipedia article, the time to launch between the two missions was 1 hour 41 minutes (15:00 and 16:41 UTC, respectively), which is pretty much 1 orbit in LEO. That would give a nodal separation of about 15 degrees. How did the two missions manage to rendezvous? I suspect the answer lays that they didn't exactly copy each other's launch trajectory, but I'm curious what the thought process was to overcome this problem.
A very detailed timeline is provided by NASA. Some of the key elements:
- Agena was in a 299 km circular orbit.
- Gemini launched 1 orbit later.
- Gemini's trajectory was such that it was in a 160x272 km orbit. This was well below Agena, and was thus moving in nodal point, although the effect was small.
- Several maneuvers were done on orbit. The first was to more circularlize the orbit. The second was to adjust the nodal plane to match, which required an 8.5 m/s burn at precisely the right point in the orbit. This was done 90 degrees to the flight path, which would have corrected the discrepancy.
- The inclination was low to begin with, the nominal out of the Cape, which allowed for the low correction to plane shift.
Once this was accomplished, they were in the same plane, and the rest is straightforward rendezvous.