There will not be any conflict created of using your certificate with the two different server. Normally you can order one certificate for your domain and install to any number of physical server.Only the thing you needed is your private key. You can get the private key from your initial server where the CSR was created and use it on the desired server.
Or you will be having an option to export the certificate as a pfx file from the server where you have installed the certificate. through a open ssl command.
openssl pkcs12 -export -in my.crt -inkey my.key -certfile my.bundle -out my.pfx
To install the certificate on the server you can separate the private key from the pfx file and use it on your new server.
Extract the Certificate and Private Key into a separate PEM file
openssl pkcs12 -in yourP12File -nocerts -out privateKey.pem
In other way some certificate authority like comodo will provide the replacement of the issued certificate with free of cost. You can check with the CA for the replacement also.So you can simply recreate a new CSR on the new server and get it reissued. So you can install the reissued certificate without any trouble.