You could perhaps show the "instant" reply right away and provide a button labeled "Think longer and give me a better answer" that starts the thinking model and eventually replaces the answer.
For this to work well, the instant reply must be truly instant and the button must always be visible and at the same position in the screen (i.e. either at the top or bottom, of the answer, scrolling such that it is also at the top or bottom of the screen), and once the thinking answer is displayed, there should be a small icon button to show the previous instant answer.
For those who are unaware, this is exactly what Grok does. The default is an auto mode, then when you ask a question it starts researching (which is visible to the user) and if it's using the expert mode but you don't really need all that jazz, it has a "Quick Answer" button right above the prom entry field, and if it's using a "Quick Answer" mode then it has "Expert" button and the same place, and you are able to toggle between them mid answer and it will adjust the model (or model parameters, I'm not sure how it works under the hood).
It's pretty good with the auto chooser, but I appreciate the manual choice available so in-your-face and especially not having it restart the query completely but rather convert the output to either Quick or Expert.
This is on the Web UI, can't speak for other harnesses. I do find that it's quite good with the citations and has a fairly generous free tier, even on Expert mode. (As for who sits at the top, I am indeed put off by Musk's clear interference in several cases involving Grok, nor do my personal values align with the majority of his, but today's Grok is definitely less MechaHitler and more reliable than it was before.)