Air carrier First Officer reported autoflight anomalies departing SAT that the reporter believes may have related to 5G interference.

Narrative

After take-off; somewhere between 1000 and 2000 ft.; I lost the flight director (First Officer side). Also missing was the airspeed bug and the FMAs on the Primary Flight Display (the thrust FMA was still displayed; but the pitch and roll ones were blank). A red 'FD2' was displayed on the Primary Flight Display. The Captain's (Pilot Flying) instruments looked normal. As our Autopilot number 1 was MEL'd; the Captain could not bring on the autopilot until I got valid information on my side. Selecting Sys 1 FD switch to display his information on my side returned the FMAs and the flight director; but still no airspeed bug; so the autopilot would not engage. Changing to Level Change (from profile climb) displayed an airspeed bug on my side. It matched the setting on the FCP; and moved with it appropriately when the speed was changed. The Captain was then able to engage Autopilot number 2. It appears the lack of airspeed bug on my side had something to do with the profile mode; although the Captain had a good display; and the information on our Flight Management System Center Display Units appeared to all match. After level-off at FL250; the Flight Management System stayed in climb mode. We tried changing the cruise altitude; going from Econ to Speed and Max Climb; to no avail. It stayed in Climb mode until sometime after we were below 10000 ft. The Captain disengaged the autopilot once we were approaching ZZZ; so I de-selected the FD1 info; and my Primary Flight Display was then displaying normally. Normally; I would not make this an safety report; but we had a very similar situation two nights before in a different jet when we departed from SAT Runway 04. In that case; the First Officer side lost the Flight Director and airspeed bug. I did not notice the missing FMAs that night; but cannot be sure. In that case; we had an operable Autopilot number 1; so the Captain took the aircraft and used Autopilot number 1. After level-off; I de-selected FD1; and my displays were normal. With a very strange; but similar occurrence on two different jets; I'm wondering if this is somehow related to 5G transmissions. From what I was able to see online; it appears there may be a Tower not far from the departure end of Runway 04 at SAT. [The cause was] unknown. Possible 5G transmission effects; or coincidental failures (and recovery) of equipment on 2 different jets on different days in a very similar fashion. I've made this report in case others are received for similar issues; [and] to help with gathering the data.