Waiver order doesn't depend on who claimed it first. The whole point of the waiver is to allow anyone to make a claim without trying to be the first (within ~24 hours of player being dropped on the wire; or until Tuesday evening if claiming over the weekend) and then claims are resolved in the reverse order of standings - last team gets first dibs, etc. One wrinkle is that when you get your claim resolved to your favor (you get the player) then you drop to the back of the line for the claim on your next player. It's somewhat similar to snake draft.
Check waiver report, this may give you some insight how it works:
https://fantasy.espn.com/football/league/waiverreport?leagueId=1730996465
Not quite, it wasn't bbc.reporter's first choice.