A sent indicator only shows that your client submitted a message; it does not necessarily mean the recipient received it. Note the local send time, the status shown afterward, and when delivery eventually appears.
Test with one harmless message to a consenting recipient. Compare web and mobile if available, and ask whether the delay affects one conversation or many. Do not send the same important message repeatedly until you know whether earlier attempts may still arrive.
Which status label is visible, and how long has it remained unchanged?