Certificates are associated with DNS/host-names, not a specific server. Certificates can therefore be copied/migrated from an old server to a new server in preparation of changing the DNS record and as part of your migration strategy.
Also consider that when you for for example use the HTTP challenge to generate/renew the certificate for api.example.com; you can't easily do that (in advance) on the new server while the DNS record for api.example.com hasn't been updated yet and still points to your old server. Migrating the existing certificates makes sense then too.
Will generating a new certificate cause any issues for my clients, especially for the native android apps? How long will clients keep the old certificate details?
Generally the server certificate gets validated by clients each time a new connection get established. (That's why usually TLS connections are kept alive for a longer time and get re-used for subsequent requests rather than establishing a new connection for each request.) Those existing connections get dropped when the DNS records update takes effect.
The main risk when switching servers is probably not in the changed certificate but in the new server using different (and more) secure cryptographic ciphers and no longer supporting legacy ciphers that older and ancient devices rely on.
