This is usually called the magnus expansion method and has a nice literature in numerical analysis. Kato also used this method to show the existence of solutions in the hyperbolic case. 

I would say that strong resolvent continuity and a sufficiently big common domain is sufficient in your case. See Section 5.3 in [Pazy][1].

I can also give a related [self-reference][2], where also the investigation of this product appears and some of the ideas are explained in a simpler situation.


**ADDED:** My answer concentrates on the method you propose to converge to the solution. To make the content of the references short: yes. A common dense domain and continuity of hte maps $t\mapsto A_tx$ implies the convergence of the product to the solution of the differential equation.

  [1]: http://books.google.hu/books/about/Semigroups_of_Linear_Operators_and_Appli.html?id=sIAyOgM4R3kC&redir_esc=y
  [2]: http://arxiv.org/abs/1105.6372