Before moving to Cloudflare, was your Website working over HTTPS connection?
If so, did you had an valid SSL certificate installed at your origin host / server which covers both your naked (root) domain any any other needed sub-domain like www, mail, etc.?
May I ask what SSL option have you got selected under the SSL/TLS tab at Cloudflare dashboard for your domain ( Flexible, Full, Full Strict … )?
Was your WordPress website configured to work over HTTPS?
Did you used www or non-www URL address to access?
Furthermore, in terms of a WordPress admin dashboard, was it working only over the SSL?