The time to shape-shift probably is related to the amount of stuff to be done, both overall quantity and the extent of the difference for a given element. Bones are the most difficult and some of them change their size substantially: the vertebrate skeleton is the same for all, just stretching and shrinking individual bones, especially so if sticking to mammals.
To work in a pinch (pun intended) he just needs thumbs. In fact, it might be handy enough (pun intended) to keep a modification to pure feline form normally, to do little things like open doors and take notes without having to change back.
You might find how to draw very interesting too
Without trying to work out and build a complete working model, you could say that the dewclaw digit is elongated and the (retractable!) Claw shaped like a tweezers tip rather than a hook; and the pisiform (E in illustration) is also extended and the end shaped as a teardrop, offering a broad anvil plus a fine tip, either of which can be used in opposition with the tweezer limb based on the claw extension mechanism for the large open/close movement, plus a little bit of movement in the finger proper to adjust the position. The tear-drop shape is wavy, offering a notch as well, with idea of being able to grasp a pen or some-such using that feature.
Without compromising his ability to use the fore paws like a Jaguar would, he can hold a pen and pick up small objects like coins. For anything more, the paw could transform much quicker than the whole body! It would offer continuous improvement to dexterity over the few minutes it takes, and his practiced trick is aimed at making the best effects from the smallest (fastest, energetically cheapest) modifications, changing the bones only slightly, curving rather than changing the length of the bones (that is the hard part), and gain dexterity at the sacrifice of leaping/clawing/running ability.
So, when faced with the need for hands, he naturally starts a well-rehearsed shift to improve his hands. Maybe opening the medikit case is a problem, being too large of a grip to work the latches; so he uses alternative means to open it; then going through the contents is no sweat, but opening the envelope packaging is like trying to open a ketchup packet with heavy gloves on and using a chopstick, so stomp on it to hold it down and claw it open instead. Grasping the tubing and needle is no problem at all, and by then his hands are becoming more useful.
Though useful in a limited sense, the sum total of manipulation abilities, strength, scale, grip etc. is all wrong so the task is awkward in interesting ways.
But, he can effect custom changes for the task at hand; changing the way a joint faces or the shape of the end of the bone without making what are actually large changes in the size of bones or anatomical features such as joint structures and tendons.
For interest, work in ways of accomplishing things that don't use his hands: maybe turning lips into knobby graspers is much faster than changing bone, and can be something kept on since it doesn't affect his feline powers (might even be an improvement without any downside).
The tail could also be used to grasp large things.