SSL certificates can't be "cancelled". They can be revoked, but that doesn't happen just like that: for Let's Encrypt to revoke your certificate, something very serious must be wrong. Like a serious incident. Changing servers isn't one of them. On the other hand, the subscriber him/herself can revoke a certificate, but that's a manual process which is almost never necessary (mostly if your private key has been leaked and isn't private any longer).
So no, your certificate won't be cancelled, just because you change host. Due not that the private key, if transfered from host A to host B, should be transfered privately. You can use
scp which uses SSH, or you can just copy/paste the contents of the private key from SSH terminal A to SSH terminal B. SSH should be private after all. Just make sure you clear your clipboard afterwards.