I'm a little bit skeptical that the chime is hardwired. The doorbell may well be hardwired for power, but if there's that kind of delay, I'm really not sure the chime is part of the circuit.

An old-school, basic doorbell is a pretty basic circuit, the kind they had me make in shop class in middle school. The doorbell just completes a circuit that delivers power to a solenoid that then hits a metal bar to make the chime.

In smart doorbell retrofit situations where the doorbell relies solely on the existing wiring for power, some doorbells can't use the chime at all, and some require a little effort in order to get them to work. The Ring doorbells require a little adapter pack that you shove into the chime its self that adds a little resistance.

It would be helpful to know what the products are.

Is your chime mounted high up on a wall? Does it never require a battery change? What are you using for the transformer that's powering the doorbell? Is it original to the home? Because it might be underpowered and maybe the doorbell is waiting to recover little bit before sending out voltage to power the chime.

It's definitely an odd situation. I have a Eufy smart doorbell that is unable to operate my home's existing chime, so I have it paired to a plug-in chime. That chime goes off less than a second after the button is pressed, and it's all going through wifi (which I hate for smart home devices but whatever).