There actually weren't that many refugees coming from what is now the North to the South. The bulk of the refugees were people feeling advancing armies from what is now the South to other areas of the South, specifically the area around the Busan (called the "Pusan Perimeter" at the time).

There were two main waves of refugees fleeing south. The first came at the start of the war when, despite repeated warnings, MacArthur and the South Koreans were caught completely unprepared by the North Korean army and South Korean and American forces retreated towards Busan quickly. This initial wave of refugees was composed of about 200,000 people and these ones were pretty much exclusively living in South Korea before the war started as any North Korean trying to escape to the South would've had to go through the advancing North Korean army which would've been nearly impossible.

The second wave of refugees came about after MacArthur and the UN forces were caught completely unprepared by the Chinese army despite repeated warnings (notice a trend here?). The Chinese pushed down a bit south of Seoul before being pushed back to around what's now the DMZ. The Chinese advance set off a second wave of refugees fleeing south of about 500,000 people, but only about 100,000 of those people were from the North.

So lots of refugees all over the place (often heading towards Busan) but only a relatively small minority of them were coming in from North Korea because very often North Korean and Chinese armies were blocking people who were in the North who wanted to go south. So people who were originally from the North wouldn't have been treated any different from any of the internal refugees in the South.

As far as what happened to those refugees it wasn't pretty. The resources of the already incompetent and corrupt South Korean government were completely exhausted by the war so you had refugee camps around Busan with food etc. being provided by the US military and various church groups. There was rampant prostitution and other black market activities as well as hyperinflation as the stability of the South Korean currency was not helped at all by the war. There was widespread malnutrition with many people reduced to eating bark etc. even outside of the refugee camps due to the disruptions of the war. If you travel around South Korea today you can still see the aftereffect of this quite clearly as the people who are old enough to have been children during the war are often visibly shorter than the people born after things settled down into the grinding poverty of the immediate post-war years which was still a huge step up from the deprivation of the war years.

Since there was so much internal displacement in the South people originally from the North would have been in the same boat as anyone else in the immediate post-war years.

As for people wanting to return North, to my knowledge that was more common among Koreans living in Japan than refugees from the North living in the South. Even to this day there are pro-North Korean groups among the Korean population in Japan that run Korean language schools, host pro-North websites, etc. The occupation of parts of the South by armies from the North wasn't pretty and as horrible as the South's military dictatorship was that was generally enough to discourage people from moving North.

Source: Kim, J. (2017). “Pusan at War: Refuge, Relief, and Resettlement in the Temporary Capital, 1950–1953”. The Journal of American-East Asian Relations, 24(2/3), 103-127.