I'm wondering why google and bing, after getting the index.html page, aren't using it to crawl the site yet.
To put everyone's mind at rest, please say explicitly that B and G have not just been requesting
index.html, but have been receiving
it (200 response).
Or rather--ahem, cough-cough--what I'd really
prefer to hear is that they have been requesting example.com/index.html and receiving a 301, followed immediately by a request for example.com/ and-that's-all. (I don't remember about bing, but last time I looked closely, Google follows redirects almost immediately--within a few minutes--unless they happen to have crawled the redirect target within the last hour or so.)
Meanwhile, what's going on with http requests? Are they separately logged? The search engines should be requesting everything on their ordinary shopping lists, and all of those requests should be receiving 301 responses to https. (301 rather than the default 302 because, well, you're not going back.) So even if for some reason they hadn't stumbled across the https site, they'd be made aware of it by new redirects.
While you're at it, please double-check robots.txt and make sure it doesn't accidentally have any Disallows that you put in there temporarily while you were making the transition.
You should expect a full top-to-bottom Google crawl within no more than 24 hours after they discover the existence of https://example.com/ And probably sooner still, if you've got a big site. (Again, I'm reporting my personal experience.) So, yeah, something is Not Right.