Ok. If this manual is the correct one, then, page 3 shows the connection pinouts on the back. If you lift it off the wall you should get access to the connections.

My guess is, what you want to try, is unplugging the SW+ SW- "door bell call button connection" plug first. Then try shorting those two pins yourself locally there. Does shorting those pins ring the chime immediately, or does it have a delay?

If it's immediate, then the "delay" issue is in the doorbell button itself or the wiring somehow. If it's delayed, then the issue is in the electronics box of the DT24 unit itself.

This doesn't SOLVE the problem but it certainly narrows it down.

While you're there, you can test for sure whether the chime rings when the pins are first shorted, or when the pins get unshorted (i.e, whether the chime is buttondown or buttonup). If the chime rings on buttonup, then your delay problem could be that the doorbell button itself is sticky (or its contacts are sticky, or there's a weak short somewhere in the wire run, perhaps the wire is submerged or rat-chewed somewhere, causing a capacitive buildup somewhere along the wire run).