Did some people expect WWII? Sure: Predictions of the next war were made as early as in 1918 (see quotes in the end of the answer). Did some people expect the war to start in 1939, applied for an immigrant status few years ahead (it was a long wait) and got their visa in time to get out before the war? Possibly, but the probability is low (since it requires a number of lucky coincidences). The probability is even lower if you want these people to have left publicly accessible letters/memoirs, recording all this. And, since you are asking for "common people," they tend not to write memoirs, so chances for somebody here coming up with a name and a reference to letters/memoirs become really small.
Here is some background, taken from the webpage Immigration to the United States in 1933-41. It deals specifically with immigration to the US.
In 1924, the United States Congress passed the Johnson-Reed Act, revising American immigration laws around individuals’ “national origins.” The act set quotas, a specific number of visas available each year for each country. The quotas, inspired in part by American proponents of eugenics, were calculated to privilege “desirable” immigrants from northern and western Europe. They limited immigrants considered less “racially desirable,” including southern and eastern European Jews. Many people born in Asia and Africa were barred from immigrating to the United States entirely on racial grounds.
The United States had no refugee policy, and American immigration laws were neither revised nor adjusted between 1933 and 1941. The Johnson-Reed Act remained in place until 1965.
Potential immigrants had to apply for one of the slots designated for their country of birth, not their country of citizenship. After Great Britain, Germany had the second highest allocation of visas: 25,957 (27,370, after Roosevelt merged the German and Austrian quotas after the Anschluss). The total allowed was approximately 153,000.
The quota was the maximum number of people who could immigrate, not a target that State Department officials tried to reach. Unused quota slots did not carry into the next year.
Requirements for Immigrating to the United States.
Most potential immigrants to the United States had to collect many types of documents in order to obtain an American immigration visa, to leave Germany, and to travel to a port of departure from Europe. Prospective applicants first registered with the consulate and then were placed on a waiting list. They could use this time to gather all the necessary documents needed to obtain a visa, which included identity paperwork, police certificates, exit and transit permissions, and a financial affidavit. Many of these papers—including the visa itself—had expiration dates. Everything needed to come together at the same time.
At the beginning of the Great Depression in 1930, President Herbert Hoover issued instructions banning immigrants “likely to become a public charge.” Immigration fell dramatically as a result. Though Franklin D. Roosevelt liberalized the instruction, many Americans continued to oppose immigration on economic grounds (that immigrants would “steal” jobs). Immigrants therefore, had to find an American sponsor who had the financial resources to guarantee they would not become burden on the state. For many immigrants, obtaining a financial sponsor was the most difficult part of the American visa process.
Potential immigrants also needed to have a valid ship ticket before receiving a visa. With the onset of war and the fear that German submarines would target passenger vessels, shipping across the Atlantic became extremely risky. Many passenger lines stopped entirely or at least reduced the number of vessels crossing the ocean, making it more difficult and expensive for refugees to find berths.
Waiting Lists and the Refugee Crisis
As the refugee crisis began in 1938, growing competition for a finite number of visas, affidavits, and travel options made immigration even more difficult. In June 1938, 139,163 people were on the waiting list for the German quota. One year later, in June 1939, the waiting list length had jumped to 309,782. A potential immigrant from Hungary applying in 1939 faced a nearly forty-year wait to immigrate to the United States.
In quota year 1939, the German quota was completely filled for the first time since 1930, with 27,370 people receiving visas. In quota year 1940, 27,355 people received visas. The fifteen unused visas were likely the result of a clerical error. It is difficult to estimate how many of these were refugees escaping Nazi persecution. Until 1943, “Hebrew” was a racial category in American immigration law. In 1939–1940, more than 50% of all immigrants to the United States identified themselves as Jewish, but this is likely a low number, since some refugees probably selected a different category (such as “German”) or did not consider themselves Jewish, even if the Nazis did.
Here is an example of a "lucky escape," taken from the book "Adventures of a Mathematician" by Stanislaw Ulam. He was fortunate in many ways, in particular, because he was already an established mathematician (most importantly, with friends and collaborators in the US) and had a string of temporary research and teaching jobs in the US in 1936-1939 (the first piece of luck). He writes:
I had to go to the American consulate in Warsaw each summer I was in Poland to apply for a new visitor's visa in order to return to the United States. Finally, the consul said to me, "Instead of coming here every summer for a new
visa, why don't you get an immigration visa?" It was lucky that I did, for just a few months later these became almost impossible to obtain.
That was the second piece of luck.
The third piece of luck, the timing and partial awareness of the upcoming war: He sailed for the US in August of 1939:
We were at sea when the announcement of the pact between Russia and Germany came over the ship's radio. In a state of strange agitation, I told Adam upon hearing the news, "This is the end of Poland." On a map in the ship's salon, I drew a line through the middle of Poland, saying, Cassandralike, "It will be divided like that." We were, to say the least, shaken and worried.
People in the United States had a much clearer and more realistic view of events than we had had in Poland. For example, when it was time to obtain an exit visa from Poland, because I was in the Polish army reserve I had to first secure the permission of the army to leave the country. The officer in charge asked casually why I wanted to go abroad and raised no further question when I told him of my lecturing engagement in America. As a rule people in Poland had not felt the imminence of war, but rather a continuation of the state of crisis,
similar to the one in Munich the year before.
But not enough clairvoyance to transfer money out of Poland before the war:
... Nevertheless we were finding ourselves in severe financial straits. The proposed income from Britain had been frozen — the English government stopped all outgoing money, and my salary as a lecturer at Harvard was hardly sufficient to put a younger brother (who was not allowed to work because he was on a student visa) through college. On previous trips, I had never thought of transferring funds or property from Poland.
Below are two examples of early predictions of the 2nd World War:
An economist Silvio Gesell (the quote is taken from here):
Already in 1918, shortly after World War I, when everybody talked about
peace and many international organizations were created
to secure that peace, Gesell published the following
warning in a letter to the editor of the newspaper "Zeitung
am Mittag" in Berlin:
"In spite of the holy promise of all people to banish war, once
and for all, in spite of the cry of millions 'Never a war again,' in
spite of all the hopes for a better future, I have this to say: If the
present monetary system, based on interest and compound
interest, remains in operation, I dare to predict today, that it will
take less than 25 years for us to have a new and even worse
Ferdinand Foch (see the discussion here):
Foch considered the Treaty of Versailles too lenient on Germany and as the Treaty was being signed on 28 June 1919, he declared: "This is not a peace. It is an armistice for twenty years". His words proved prophetic: the Second World War started twenty years and 64 days later.
In both cases, the timing (20-25 years) was surprisingly accurate.