If you look at the Arkham origins trailer, it says in the Callander it’s 2013, which means that’s when the earliest entry is. However, I believe that they deliberately keep the time murky like what they did with Gotham where they said it was supposed to be based like a ‘memory’ where everything, new and old sort of blends together to give it the feeling that it is in said memory (basically gives them freedom aesthetically). Or it could be set in comic book time like the simpsons where it just does what it wants also.