Re: delayed mechanism
Getting the tension and timing just right on the body octave mechanism is one of the tricker areas of fine tuning a horn.
As mentioned above, congealed or dirt laden gunk in the hinge tubing and post joints is one possibility, worn out or maladjusted springs is another.
One not mentioned yet, but also possible is a bent hinge tube and or hinge rod.
Although I work on and restore horns myself, and would have no hesitation fixing this personally, if you are not skilled and experienced in the tech side of horn ownership, I wouldn't recommend disassembling the mechanism yourself. Get the spring tension wrong, loose a cork foot etc. and your problems will be exacerbated.
This can usually be put right quite easily and inexpensively by a good tech, and that is who you should turn to.