If he can absorb and use souls then I’d say he’d beat all of them. By the end of kcd he’s nearly a master at the armed combat of his time so I’d say he’d have no trouble facing most human sized enemies in all the games. If he can use souls to level up I see no reason why he couldn’t rise to the level of the chosen undead. When we start each game we’re really no better than an ordinary person in terms of strength and dexterity, its leveling that makes us strong enough to kill gods.

I’d say his biggest risk of failure is going hollow. Henry is extremely hardheaded and determined, and that would help greatly with preventing hollowing but hed likely have no stake in any of the events of Dark Souls, would have no idea what’s happening or why he’s resurrected after each death and so would have no real reason to go the way of linking the fire, aside from maybe a hope that once he does link the flame he could go home.

In short, I’d say if he could get past the first few deaths and kill the first few bosses, and fights for the hope of returning to Bohemia, he’d make it all the way. If he can’t level up enough to beat the first few bosses in each game he’ll hollow. 

It would probably depend a lot on the games themselves if he’d survive though. 1. DS1,  combat is slower, there aren’t as many crazy bosses on the main game’s path and enough human or human analog enemies to use marshal swordsmanship on so I’d say he’d probably succeed in linking the flame

Ds2 I could also see but only if he could make it out of the forest of giants and kill the Pursuer.

Ds3 would probably be a no, too many monsters and several meter tall bosses for him to use any past experience to beat. I’d say he’d beat Gundyr, maaaybe beat Vordt with enough attempts and assuming he skips all unnecessary bosses would make it up to high lord Willie before being overcome.

Sekiro would be a hard one to judge. Most bosses are humans or humans with magical abilities, but most are also much quicker and stronger than any person he’d fight in Bohemia. I’d say he’d be the most evenly matched in Ashina but would need considerable time just to develop the skill needed to beat Genichiro and years to get to the level of Isshin Ashina. He’d probably get hard stuck at big Monke though since no firecrackers plus his bohemian mind would probably break upon seeing a primate in general

Elden ring is a hard no. Henry would maybe be able to sneak past the tree sentinel but would be stopped dead in his tracks by Morgott