Global 6000 Captain reported track and altitude deviations departing DCA; citing improper autoflight programming and a possible wake turbulence encounter as contributing.

Narrative

REBL5 departure Runway 1 Selected NAV for take off. Briefed RNAV to REVGE up to 5000" Normal takeoff. At Flap Retraction Altitude I called for flight level change. I didn't notice the Pilot Monitoring (PM) had selected HDG instead. The plane didn't turn left over the river and we were headed straight for P56A. I disengaged the autopilot and made a hard left turn to stay over the river just as we entered the clouds. I also wanted to reduce my speed in order to make the turn tighter so I disengaged the auto throttles. I then noticed that we were not in NAV so I re-selected it. There are many fixes in close proximity there so all I could do was follow the lines on my FMS and keep the plane on the lines. Losing altitude during one of the turns we stopped climbing and I heard "DON'T SINK" followed by "GEAR". I think the GEAR call may have been caused by a RA 1 failure that also was annunciation at this time. PM later claimed he thought we had hit wake turbulence in that first turn and reported it to ATC. They asked if we were ok and commented that we were climbing very slowly. Once we were away from the prohibited areas and on a fairly long straight stretch of the departure I re-engaged the AP and Autothrottles and we resumed climbing normally. Be extra careful with inexperienced PM's and have a plan for unexpected improper auto flight mode selections."