Thank you, OP, had been wanting for months to know about this issue. Though I did everything you said, even had a new WP installation on a old and unused domain I have, and having done the changes to my GoDaddy's account's DNS settings more than 36 hours ago, I am still facing some problems.
At present, after about 36 hours, I can see my https WordPress site as it should, but when I do a /wp-admin, the error I get is:
CHROME:
Error message (on Chrome):
This webpage has a redirect loop
ERR_TOO_MANY_REDIRECTS
The webpage at [https://mysite.com/wp-admin] has resulted in too many redirects. Clearing your cookies for this site or allowing third-party cookies may fix the problem. If not, it is possibly a server configuration issue and not a problem with your computer.
I am looking for solution on Google right now, and am hopeful to find it.
FIREFOX:
Also, in Firefox where I started doing the changes, there too the error message is:
Error message:
The page isn't redirecting properly
Firefox has detected that the server is redirecting the request for this address in a way that will never complete.
This problem can sometimes be caused by disabling or refusing to accept cookies.
CMD window:
I have done the ipconfig/flushdns, ipconfig/release, ipconfig/renew etc many times in this period.
CLOUDFLARE:
The DNS Settings on cloudflare account:
Qt.1:
The DNS still shows the old ip address (of the old host this site's DNS settings were set to). Shouldn't it show the new (cloudflare IP address) by now, after 36 hours?
I have set the cloudflare icons to orange too.
GoDaddy:
The GoDaddy account's DNS new details show that the nameservers are of cloudflare's.
My Cpanel:
Addon domain details show that the said domain is not redirected.
Qt.2:
Now how long do I have to wait for that problem to resolve if the issue is on server side, and not a problem on my PC?
And what can I do in the meanwhile, or if the issue isn't solved even after 48 hours (another 12 hours to go)?
The WP site was installed new, and now I cannot add any plugins to it as I cannot log in to the wp-admin...
Will keep reading, try finding, and hoping that everything settles smoothly. The https means a lot to my son who has some unfinished business with his Android app completing his 1st $100 on Adsense. Of course he is earning a lot through his sw-development+web-designing job, and through freelance jobs, but getting the first Google Adsense cheque is another feather in his cap!
He said (sometime back) that he needs the https on his site to get his Android app working again... It breaks my heart to see him waiting eternally for a working solution, and I feel that OP has provided it at last. Just a matter of few hours now.
TY to OP, and Good luck to all others!