AdWords API does not have the start time, nor any related info like the time the first user was added, or the time that the first user agreed with Terms & Conditions, or anything like that.
You may want to consider what you really need. In my case, every now and then I get a new account which is very large - so large you cannot just ask for reports, segmented by date, with ALL_TIME: you get an error. To overcome that, you ask multiple reports in periods. But what would be a suitable start time? Just starting with January 1st 2000, and asking 30-day reports is not that optimal. (Plus, certain reports of me give an error on 30 day periods, sometimes I have to get down to 4 day periods!)
For my case, finding the ealiest start date of its campaigns is good enough. Before the first start date no activity can possibly happen. To get the first start date in AdWords API, use campaign service, have the results sorted by campaign start date, and limit the request to one - no need to get multiple campaigns. Also make sure that deleted campaigns are not filtered out, as they can be the oldest campaign.
In Ads API things work different but you get the idea.
Again, this method works for finding the earliest date on which activity can be reported. You may have a different use case.