If they arrived (but with no immigrant arrival record*) in the 1940’s they might be eligible for Registry/Lawful Entry proceedings after 1965 or 1986. Let me explain.
This question highlights how Registry proceedings were always “after the fact.” The 1929 Act called it “Registry,” but by the early 1940’s INS changed the name to “Lawful Entry”—but it was the same legal provision, and remained the same provision in the law to this day.
The 1929 Act set an arrival “cut-off date” for eligibility. This means under the 1929 Act only people who arrived prior to June 3, 1921, could apply for Registry. Over the years Congress updated the law, moving the date forward. Whenever they moved the date forward additional people (later arrivals) became eligible for this form of “legalization.” The changes were:
1929 (45 STAT 1512) Eligible if entered (arrived) prior to June 3, 1921
1939 (53 STAT 1243) Eligible if entered (arrived) prior to July 1, 1924
1952 (66 STAT 219) Eligible if entered (arrived) prior to July 1, 1924
1958 (Pub. L. 85–616) Eligible if entered (arrived) prior to June 28, 1940
1965 (Pub. L. 89–236) Eligible if entered (arrived) prior to June 30, 1948
1986 (Pub. L. 99–603, as amended by Pub. L. 100–525) Eligible if entered (arrived) prior to January 1, 1972
Thus if someone arrived in 1945 but without an immigrant arrival record*, they would not become eligible for Registry/Lawful Entry proceedings until 1965. If they arrived in 1949 they would not be eligible until 1986.
ALSO, the USCIS Registry Files only hold this material between 1929 and 1944. Beginning April 1, 1944, all applications for Registry/Lawful Entry were filed in an A-file. And if the person naturalized prior to April 1, 1956, all that material would have moved to the C-file.
Sorry it is so complicated. I hope this helps,
*You might find someone recorded in the records as admitted temporarily (non-immigrant arrival, or crewman record). But that is not a record of admission as an immigrant, so they have no record of LPR admission.