Negligible influence. This idea that Christianity can win converts and have an impact "if only it wasn't persecuted!" is a horrible concept and plain wrong and assumes Christianity is superior.
Christianity was winning converts. While in the very early days some peasants thought they were joining a weird form of Buddhism (Monks, prayer beads, etc). To deny that is to deny reality. Yes, Christianity used violence to gain the upper hand a lot of places historically. But Japan isn't one of them.
But look at the competition they had at the time. Shinto was subordinate to Buddhism, and Buddhism, other than the warrior monks protecting their own domains and the peasants on them, were very subordinate to Daimyo and upper class. Being on the government payroll like they were in the Tokugawa didn't inspire the monks to be inspiring.
This did play a part though in the revival of Shinto in the later years of the Tokugawa (one that didn't end until the power of Amatarasu Okami was unleashed on Hiroshima and Nagasaki) and it's embrace by anti Tokugawa forces that led to the Meiji restoration.
All of this gives me an idea.
What if Christianity manages not to anger the powers that be in the 1600's and survives. Anti Tokugawa forces, particularly in the South, turn to it or partially it or to some weird Shinto/Christian Hybrid (rather like Voudom, only Japanese).
That could be interesting and fun.