Preventing replication of the signature of the JWT relies on you storing the key correctly that signed the JWT, either in a secure storage or some other safe place, reuse is a different story.
Now, for a start, I would also be worried if an attacker can easily steal your JWT, which almost implies that you don't use HTTPS/SSL, but sure, let's assume we are running a scenario where someone has exploited your SSL.
JWT's themselves are nothing more than ways of authentication a client, almost similar to how an API key works. It's pretty much a grant that that client can access some resource using that token. It doesn't hold any security value in that regard, other than at some point that client authenticated itself using the correct credentials.
So if it's stolen, you have a problem. This is where the lifetime of the tokens comes in to play. Tokens can expire very quickly, causing the attacker to only be able to access resources for a short while, but still, they did have access to a resource they shouldn't have access to.
In that regard, you can try to mitigate the attack using, like you mentioned, a nonce. Just don't forget to protect that nonce by a key that is related to the identity or session of the JWT. Alternatively you can add an HOTP or TOTP, or some unique signature to the request. This comes at a cost of you having to create extra logic at both your client and server.
So summarised, JWT's themselves don't really provide security, they provide authentication. Generally speaking JWT's are secure if using contemporary security measures such as SSL. If you really want to add additional security to the request, you are left to implement additional logic, such as OTP's or signatures by implementing more logic at the client and server.