It's not too logistically complicated on its own. You just stop takeoffs, start taxiing the conga line of waiting airplanes to the new runway, let airplanes currently on final land, and start vectoring the other airborne flights toward the new approaches once the last of takeoff traffic is away. Once the new landing flow is established, start releasing the takeoffs. This will require coordination between tower and approach and may involve the center if flights need to be put in holding during the transition (depends on how busy the airspace is). In certain busy airspace this may also require coordination with other nearby airports.
With that said, some airports that take off and land on different runways can have impacts on arrival rates in different configurations, and this adds to the logistic complication. One example is KIAH, which in normal flow is taking off on the 15's and landing on the 26's and 27. It isn't too much trouble to switch landing traffic to the 8's and 9, but if takeoffs have to be switched to 33 then the flow rates are negatively impacted. It gets even worse if landing and takeoff traffic have to be put on the same runways. How the arrival and departure flows are impacted (if at all) from a runway switch will depend strongly on the specific geometry of the runways.
Anecdotally, the logistic impact of reduced arrival and departure flow is probably the largest impact. At KIAH they will run their normal flow until the wind gets so bad that pilots start refusing clearances for tailwind limitations, and only then will they switch the departures to northbound. Similarly, as soon as the tailwind is legal, they'll switch back to southbound departures.