The launcher itself shows nothing
This surprises people coming from Steam, but the Battle.net desktop app is a downloader and patcher. It lists your games, it does not report how long you have played them. There is no hours column, no per-game total, and no profile page that sums your time across Blizzard titles.
Everything below is per game, and each one works differently.
Where each Blizzard game keeps the number
-
World of Warcraft. Type
/playedin chat. It reports total time on the current character and time at the current level. There is no account-wide total in the client; you check character by character. - Overwatch. Career profile statistics include time played, broken down by hero and by mode. The presentation has changed across versions and seasons, and historical data has been reset in the past.
- Diablo IV. Character screens show time played per character rather than an account total.
- Hearthstone, StarCraft II, Heroes of the Storm. No reliable in-client playtime figure. Games played and ranked records exist, hours generally do not.
If a game you play is not in that list, assume the number is not exposed. Blizzard has never treated total playtime as a headline statistic the way Valve does.
Why per-character totals are not what you wanted
A /played figure answers "how long has this character
existed in play", which is a different question from "how much of my
life went into this game". Anyone with several characters, or with
years across an account, has to add numbers up manually and still
misses time on deleted characters and other Blizzard titles entirely.
One consistent number for every Blizzard game
PlayCounter times the Windows process. Launch World of Warcraft, Overwatch, or Diablo IV however you normally do, and each gets a session recorded at the game level, with a running total and a session count. It works the same way for the non-Blizzard games on the PC, so you can compare them directly.
It measures real elapsed time with the game running, so it counts queueing, menu time, and idling in a capital city - the same behaviour as every store counter. And because it identifies games by executable, it keeps working when Blizzard reorganises a client or resets its own statistics.