I finished episode 2.
But I think this storyline feels like something that could have been in the 2005 season. Eccleston sounds fine to me. I think they got the dialogue of the character right.
I wasn't expecting the episodes to be connected to each other but actually this enhances the story and makes it grander. It also makes it feel shorter in a way.
I'm on the second episode and I actually like that he jumped a bit around his own timeline. I like that he crossed his own timeline by mistake and caused the whole mess. Kind of like The 11th Doctor's first season.
I wasn't expecting the episodes to be connected to each other but actually this enhances the story and makes it grander. It also makes it feel shorter in a way.