You can’t use your certificate for your visitors to see unless you are on the Business plan or higher where you can upload a custom cert.
The certificate on your server is still required though, as you need to use Full (strict) to make it fully secure. The cert on the server is needed for Cloudflare to connect securely.
Your visitors will see the Cloudflare certificate to connect there, but Cloudflare will use the certificate on your server to connnect securely there.
On the free plan, the certificates can take up to 24hrs to issue, but normally it is less than that, on the higher plans I believe they issue in around 15 minutes. The cert will only start issuing after you change the nameservers and this is detected by Cloudflare. I would recommend setting all your DNS entries to to keep your site functioning as normal. Once the cert is issued, you can set to to enable all the Cloudflare services.