To avoid downtime while changing DNS server, you will have to set up the same DNS records on the new server first.
To do that in DH, add the domain to your hosting plan first and set up DNS records accordingly via DH panel --> Domains --> Manage Domains
For your case, you will have to point MX records to gmail. Once you have the DNS records ready in DH, it is safe to change the DNS server. Whether the request reaches the old or new DNS server, both will forward it to google correctly.
If you are not sure, I’ll suggest you to test the above process with a subdomain first.