I'm a huge fan of hanafuda/hwatu and collect the cards (including an old paper Nintendo deck). I'm glad to see others who are interested.
A prohibition of gambling has been a common feature of Japan's history since the 1800s, which extended to their colonies as well. There was a Portuguese game from which hanafuda was developed, in large part as a response to a prohibition^1. The cards became hugely popular due to the craftsmanship and design, but not enough to bring in huge profits.
With gambling being taxed in the period before the war and restricted more heavily during the war, it wasn't really a boom time for the company. Much like Disney did in the time in the US, Nintendo produced cards as patriotic propaganda, with re-written versions of classical poetry. Here's a photo of one such set. People still played hanafuda and other than these additional pro-Japanese designs coming out (which sustained business long enough to keep the company from collapsing), things didn't really change for the company. The economy was bad, but they survived.
It wasn't really until Hiroshi Yamauchi, the adoptive grandson of the adopted son of the original founder, that much changed. He was kinda a jerk, but he was also the guy that eventually pushed the company into electronic gaming. Before that he expanded more into western playing cards, which you can still find on ebay. Note the two 壹圓 stamps on the back in the last photo. That's the 2 yen gambling tax in effect. You'll find those stamps on all the cards from the period.
tl;dr: Other than economic problems that plagued most companies in Japan at the time, and the need to redirect some efforts toward propaganda^2, the war didn't have a major effect on Nintendo.
This didn't work well long term because eventually hanafuda cards were also seen as primarily a tool for degenerates to gamble (as is still the case today in the region).
Also pretty common around the world at the time.