The invasion proceeds very badly. For everyone.
Softening up for the invasion with bombing and bombardment leads to massive casualties among civilians. Like, appalling.
However, it really doesn't affect what the Japanese military forces have become, because....
1. They don't have any real infrastructure upon which they depend;
2. They don't have any equipment that's critical;
3. Except for a few of them who are hit like...DIRECTLY by bombardment and literally blown to pieces, they're fine afterwards.
After which the invasion begins, and then to their horror the Americans find themselves against thousands of nigh-unstoppable warriors who move too fast to follow. Bullets that actually hit them don't stop them, and barely slow them down (unless it's a massive fusillade directed at one person, and that leaves a whole lot more Wolverine Soldiers not being fired upon). Then they get up after they're 'killed' and stab their killers in the back.
They also very quickly arm themselves with the gear from the invasion force that makes it onto the Japanese islands, and things get even worse.
Aside: The adamantium swords add little to the Wolverine Soldiers other than basically giving them an unbreakable weapon..
Anyway, the invasion fails with truly massive loss of life on both sides (except for the Wolverine Soldiers, who come out of it with minimal losses). The Americans go back to bombing Japan. The Japanese civilians suffer and die. The Wolverine Soldiers basically become an undefeatable guerilla force who cannot affect the world in a meaningful way, because they can't really leave Japan.
Tie game, everyone loses; this is really a horrific scenario.
Alternately, without the atomic bomb being dropped, Japan manages to work out a negotiated peace before the invasion begins, as according to many
it was already preparing and/or attempting to do. <-- I like this version a whole lot better, even if it means we have thousands of people with ridiculous healing factors running around. Certainly the human race would look very different by 2022!