About 1 year to return to a "new normal". 20-30 years to replenish the population so we can call it "full return".
While losing half of the population seems to be a devastating blow, there is nothing that would prevent the world from quickly readjusting. There are a few things that would change the balance, like major natural disaster which would coinciding with disappearance event, or triggering of the nuclear war. Barring those calamities, things should stabilize pretty quickly.
The main key is that all emergency services - army, law enforcement, firefighters, hospitals, major utilities - are designed with redundancy and flexibility in mind. Losing half of the staff in a military unit or police department will be tough, but those institutions should quickly readjust and try to do their best. Hospitals will also stay open, and lights will stay on.
Transportation would not take this event easy. Airlines and airports will have to shot down and rethink what to do next. But buses can keep running.
There will be plenty of food supplies in warehouses and fuel in the depots, so there would not be any risks of starvation. Some supermarkets will close, but army and national guard will keep the worst off.
On day 1 there will be a lot of chaos, but it would not develop into an apocalypse. Police will be out on the streets, electricity and phones will keep working, and food will be easy to find.
There will be a need to quickly find all the children that were left without parents and ill/disabled/old people who were relying on their relatives for survival.
What is the biggest risk is that government institutions that were not designed for emergency operations would be paralyzed. For example, in the US, if half of Congress, half of federal department's staff, half of the Supreme Court (give or take a person, because there are 9 judges), and maybe the President himself disappeared, it might be difficult to show the necessary leadership and steer the country out of crisis.
For the first month or two the crisis would deepen, however, there still not going to be break down of order, no massive riots, no civil war, and hopefully no international war as well.
After the government will find its footing and people would accept the new order, things will start to improve. In the new economy there will be half as many workers, but then there is only a half of population to support. Planes will start flying on a new schedule, schools will reopen (though not all of them), factories will resume production. In one year's time, the world would be on a firm path to recovery.