Hi Conor,
- Is this the default JIRA mail server or your own mail server?
If you're using Exchange or Office 365 take a look at How to prevent Outlook from adding signatures:
Your users must be logged on to a domain to which you have full administrative access as they must be subjected to your GPO policies for this solution to work. Organizations working with Exchange Online (Office 365) can also apply this solution if their users are connected to an on-premises Active Directory or Azure AD to which you have full administrative access.
Another option may be an add-on such as the Enterprise Email Handler for Jira. They have an article titled Remove Signature Text and Images that may help you do what you want as well.
Hopefully that helps.
Cheers,
Branden
