Nothing, the other site you shared is not using cloudflare and will not be affected. Suggest you continue with the nameserver change for this zone that is pending
The other site is using ns1.translatum.gr and ns2.translatum.gr as DNS. If I change the DNS for translatum.gr to Cloudflare DNS, how will the other site function?
As long as you continue to operate the ns1 and ns2 nameservers in translatum.gr and set their hostnames to DNS Only you will be fine. If you are planning on decommissioning those nameservers, you will want to move the DNS for the hosted zones first.
I use Directadmin so I guess nameservers are set up there.
set their hostnames to Cloudflare DNS Only
You mean simply implement the change at registrar level to Cloudflare DNS for translatum.gr and leave everything else as is?