Once you switch to Cloudflare, Cloudflare becomes your DNS. DNS entries refresh frequently, so a DNS change here should take effect within 5 minutes.
As long as you have the proper IP address in your Cloudflare DNS “A” record, that’s all you need at this end. Then it’s up to your host configuration to serve the proper site at that IP address.
So if your old host had your site at 188.8.131.52, then you switch and update your Cloudflare DNS entry to use 184.108.40.206, the switch should happen within five minutes.
Sometimes, to troubleshoot, it’s handy to the entry to untangle any issues you have with hosting before you turn Cloudflare on. Just make sure your DNS entry has a TTL set to 5 minutes or so.