The plot is certainly all right overall I feel - nice way for them to meet up, and character-wise they seem pretty accurate (I don't know Layton characters that well but they seem to be right at any rate, and the AA ones look good to me). I did think that the characters sleeping at the end of the last scene felt a bit off though, especially given for instance Phoenix often not getting sleep when he has a case. Maya in particular seemed a bit too casual in proclaiming that she is going to sleep imo.
There's still the grammatical mistakes here and there as well, which I suppose would be another reason to take your time with writing/edits. A good method I find is to, after writing it all, give yourself a day's break from looking at your story so when you do re-read you have a fresh look on things, and it's easier to spot your own errors or see what parts need work. Here's some of them:
"Now Luke, quite down, we must let Flora get her rest." Professor Layton responded to Luke's uproar, motioning to a sleeping Flora in the back seat.
Besides that it should be quiet, there should be no full stop there after 'rest' as the part following the dialogue talks about who spoke it and flows on, so it ought to be treated as one sentence with the dialogue. Hence no full stop, but a comma instead (as otherwise it's like putting a full. stop in the middle of a sentence - effectively you ignore the quotation marks).
"Hmm... I don't think we will arrive in London before nightfall. I believe we must find a resting place for the night." Layton concluded, noticing the darkening sky ahead.
"Maya, I don't think there are any waterfalls here, stop asking already!" The man said, addressing one of the robed women.
Same here - full stop should be a comma, and 'The' should be 'the'. There's some other instances of this as well, along with times you did it correctly as well, so just look through your work to fix those errors.