Horace Mann went to Europe to study education systems after he had already been appointed secretary of the just-founded Massachusetts Board of Education. He had already published his mission statement, his guiding principles etc. -- his fact-finding mission to Europe certainly gave him a lot of practical insight, but he was mostly confirming the validity of the ideas he had already worked out. One may even say that, to some extent, he didn't so much learn from the European examples as he simply used them to justify what he had already decided upon.
As such, if the intent here is to mainly explore the effects on Mann and his work in the USA... that's going to be quite minimal. The effects on Prussia will be considerably greater, of course. The Prussian education system laid the groundwork for a relatively broad educated class in Prussia, which greatly contributed to Prussian (and German) achievements in various fields. After all, own-bias-confirming as Mann may have been, he wasn't wrong when he pointed out the merits. There were other countries who implemented similar reforms, of course, and there position compared to Prussia might improve. Germany may less obviously be the great leader in science and engineering in this ATL.