Unfortunately North Korea itself is not my precise area of focus, and a lot of the pertinent research is not available in the languages I am familiar with, but hopefully I can give you a broader, more general answer until hopefully someone better qualified than me comes along.

But in basic terms - yes, many of North Korea's Christians chose to flee to the south of the demarcation line soon after the Soviets have established a consistent presence in what is now North Korea.

The reforms that started in 1946, soon after the Soviets arrived (and were almost entirely driven by the Soviet military commanders on the ground and the Soviet embassy rather than North Koreans themselves) entailed a major redistribution of land and nationalization of all industries, which is when the majority of North Korea's land-owning and entrepreneurial classes chose to flee.

Around the same time the newly established regime embarked on a campaign of political persecution of all those who might oppose it (and it first people were even being sent to Soviet camps, as North Korea's own detention system was still under construction). This included both local nationalists, internationalist communists, and many others, but also Christian activists.

Same with landlords, it was during that period (~1946 to 1951) that many Christians likely crossed the still poorly guarded demarcation line into South Korea.

We don't have the exact figures but the estimates usually fall in the region of 1.2-1.5 million people leaving North Korea during that period, amounting to between 10 and 15% of the country's entire population at the time, and it's safe to assume that a large proportion of North Korea's Christians (or at the very least Christian activists) either left the country, or faced political persecution during that time.

After the Soviet withdrew much of their occupation force around 1949, the North Korean persecution of Christians and other religious groups continued and, if anything, became more ferocious - for example targeting and imprisoning the few remaining foreign missionaries and priests, who were allowed to continue monastic practices under the Soviet rule (such as in Tokwon Abbey).

Unfortunately, a more detailed answer is very tricky to give due to the secrecy of the regime and my own limited knowledge and ability to access accounts in Korean. Ironically Kim Il Sung's father was himself a Christian activist, and a good chunk of North Korean leadership at the time also had Christian familial connections in one way or another.

Records suggest that limited Christian activity was allowed to resume under the strict control of the state throughout the 50s and 60s, and defector accounts suggest that a crypto-Christian movement, effectively underground practice, also existed. The so-called Songbun system of ascribed status that was put in place by the mid-60s, and was essentially a caste system, divided the population into 3 large groups (with further sub-groups) largely based on their family history - basically "hostile", "wavering", and "loyal". It is known that descendants of known Christian and Buddhist priests and activists were considered to be part of the "hostile" group. That did not necessarily lead to direct persecution, but severely limited one's options and entailed a large degree of discrimination.

There are conflicting accounts from defectors as to the degree to which one can openly practice religion in North Korea today (ranging basically from restrictive to completely underground), and there exists an official state-approved and run body (such as the Korean Christian Federation) representing the country's Christian population, and we know that several new Protestant churches were constructed in the late 1980s and early 1990s. In 2002 as part of the report to the UN Human Rights Committee, the North Korean government stated that there were 12,000 Protestants, 10,000 Buddhists, and 800 Roman Catholics in the country, which suggests that the proportion of religious adherents (not just Christians) among the population dropped from about 24% in 1950 to 0.016% in 2002 which does not account for whatever underground activity that is likely still going on. We also know that several Russian Orthodox Church priests are present in Pyangyang in the country's only Orthodox Church that (at least initially) mostly served the needs of Russian embassy officials and other foreign delegations.

But to give a shorter answer - it is likely that a large portion (if not most) of North Korea's pre-1945 Christian population escaped to the South between 1946 and early 1950s. The remaining Christians faced political persecution and were likely forced to either cease all religious activities, or take them underground. While there are signs that the limits on religious practices have been loosened over the last 30 years, it is no way to suggest that there is freedom of religion in North Korea today.

For general information I mostly relied on Andrei Lankov's "The Real North Korea". Another source that I consulted at the time was Charles K. Armstrong's "Tyranny of the Weak: North Korea and the World, 1950-1992", but I hesitate to fully recommend it on account of the issues with plagiarism.