This is really a great start to your story.
I know you may be a little stuck with where to go from here, so if you don't mind, here are some ideas you can think about.
o - Lily says she doesn't know who James Potter is - Harry should do a little looking to see if Voldemort's spell didn't move him to an alternate universe. If that is the case, then this Lily Evens is not his mother and all is right with the world. What he does with Petunia is to prevent her from getting hooked up with Dursley at all cost.
o - In this reality, is there:
Voldemort?
Dumbledore and Hogwarts?
What of the Goblins?
Weasleys, Blacks, Tonks etc.?
o - What are some of the challanges facing the wizarding world?
o - How can Harry get Lily and Petunia both feeling comfortable with magic. Could Harry help Petunia 'awaken' something within herself to make it so she doesn't feel left out?
o - How will Harry interact with his 'Grand Parents'? Even if they aren't 'his', they still would be a close representation of what he would have had in his other world.
There is too much potential in this story to just let it sit by. Maybe the 'Long Road Home' is Harry finally finding a place where he can be at peace with himself and his loved ones. He doesn't have to 'get back' to his original time-line, it doesn't exist any more.
He really should talk the old lady's advice and just move ahead from where he's at now.
I look forward to your next chapter.
rdgale