A321 First Officer reported a last-minute runway change issued by ATC during taxi led to a track deviation. The flight crew also encountered periodic wake turbulence on climb-out from a preceding heavy aircraft.

Narrative

I was the FO on Aircraft X (LAX - ZZZ). We departed Gate XX and were cleared from the alley to Runway 25R via Taxiway B. Our departure clearance was to climb via the ZILLI 5 except maintain 5000 feet MSL. Shortly after departing the alley; our taxi clearance was amended to taxi to Runway 24L via the City Ramp; specifically taxi Lima; hold short of Taxiway K1. Following contact with City Ramp we're cleared to taxi Kilo 1; Kilo; and to contact Ground Control holding short of Taxiway Delta on Kilo. During this portion of the taxi; I changed the departure runway from 25R to 24L and then verified that the correct points for the SID were displayed and that there was no flight plan discontinuity. I then updated the takeoff performance data; noting that the performance number for Runway 24L were in fact the same at 25R. Once approaching Delta taxiway; Ground cleared us to taxi to Runway 24L via Echo taxiway. During our taxi on Echo; we were switched to Tower Control and Tower asked if we had the numbers for an intersection departure. I told Tower that we did not have the numbers and they cleared us to taxi for a full-length departure from Runway 24L. When given our takeoff clearance (RNAV to DLREY; cleared for takeoff Runway 24L); I again verified that DLREY was the correct fix displayed on the SID in the FD Pro application and that DLREY was displayed on the Navigation Display (ND) and in the flight plan. Takeoff was uneventful but we did encounter periodic wake turbulence on climb-out from a heavy aircraft that departed before us. Upon checking in with SoCal Departure; we were cleared to climb via the SID. At this point; we had not yet reached 5000 feet. After we began a left-hand turn from NAANC to TANDY on the ZILLI 5; SoCal Departure amended our level-off altitude to 6000 feet and gave us right hand turn to 250 degrees. Shortly after; we were given clearance to our next fix.During the flight; the CA and I discussed what may have been done incorrectly; but the conversation was inconclusive. Additionally; I went to the secondary flight plan (which was a copy of the primary flight plan) and switched the runway from 25R to 24L to determine if there was a system issue that I had previously overlooked. When I changed the runway in the secondary flight plan; the departure waypoints cycled from DOCKR; WEILER; and SHAEF to DLREY; ENNEY; and NAANC; as expected for Runway 24L. After the flight; I reviewed the departure and it showed the aircraft maintain a heading of about 253 degrees for approximately two minutes and twenty seconds (2:20). Additionally; it shows the aircraft began a left turn beyond 10 NM; which is consistent with the turn from NAANC to TANDY on the ZILLI 5 SID.A runway change during taxi is always challenging. Slow taxi or pulling over to ensure all departure and takeoff data is updated and correct is a must; even if it means briefly delaying departure.