The other Mass Effect wiki has a timeline which simply says the events of ME1 (that is, the "Eden Prime war") take place in 2183CE. There are other clues in-game and in published material (like trailers) which may offer other clues; for example, Shepard's birth date is April 11th., 2154CE as attested both by a trailer and by a developer's tweet.
Another bit of information harder to source is that Shepard is 29 when ME1 starts - hence, ME1 begins no earlier than April 11th. 2183CE. As to the end date, we don't know a lot. There is that opening scene of ME2 where Shepard is killed, one month after the events of ME1.
There are likewise a few other scattered clues, such as Jacob's assertion that it took two years and twelve days for the Lazarus Project to revive Shepard; or Tali's remark (in ME3) that just three years previously she'd been shot at by Saren's agents on the Presidium ring, an event very close to the start of ME1.
Working back from such dates known for the later games, my best guess is that ME1 ends on or about 5th. October 2183, with Shepard dying the first time a month later in early November 2183.
Your mileage may vary.