Hi iTomaz,
Perhaps we can isolate this a bit further. Does this happen for iMessage (blue) as well as for SMS (green) or impact a particular message type? If you try using the same tones for other alert types, what happens there? For example, try setting this as your mail tone from Settings > Notifications > Mail.
Since you also mentioned an Apple Watch, the following information from Notifications on your Apple Watch - Apple Support helps explain when and where notifications should be available.
"If your iPhone is unlocked, you get notifications on your iPhone instead of your Apple Watch.
If your iPhone is locked or asleep, you get notifications on your Apple Watch, unless your Apple Watch is locked."
All the best.