I had the same problem and spent hours researching trying to link it and after contacting support and a long migraine I have the answer.
First step is to login in the mobile app and check the little box that says “Link Account” if the box says that it means it’s not linked to a email.
If it says “view inbox” then that means it’s been linked to a email. Which is good
Now I’m gonna paste what I got emailed from support;
Seems like you have a Jagex Account and a RuneScape Account with similar login info. Jagex Accounts and RuneScape Accounts use different links/methods for recoveries. So first of all, it's important that you recover your RuneScape Account, by following the steps below:
Head to this page Enter in your login email Click the 'RECOVER' button This will send you out a password reset email, so head over to your emails inbox and look out for it arriving and then click the link within the email to set a new password and you'll be good to go!
Be sure to check your spam/junk folders in case you don't find it in your main inbox, and if you are unable to locate the email checkout this page where we have some advice on how to proceed next.
Remember to NEVER share your password anywhere else, including with Jagex via ticket or email.
Once you have recovered your RuneScape Account, you'll be able to import it into your Jagex Account by following the steps below:
Log into your Jagex Account at https://account.jagex.com/en-GB/manage byentering your Jagex Account email address and your password Head to the "Characters" pane along the top of the page
3. Click "Import Existing".
4. From here, you will need to log into your character which you'd like to import. Please bear in mind you will need to log in using a login email + password, not a third party login association like Google or Steam.
5. After you have logged in, you'll be able to import your character into your Jagex Account.
Please bear in mind that once your character has been imported into a Jagex Account, you will only be able to log into it through our Launcher.
You will need to make sure you have logged out of the Launcher first, then log into your Jagex Account via the Launcher using your Jagex Account email and password. From November 7th, only Jagex accounts will be able to access THAT Jagex Launcher.
Once you have done that, you will be able to select the character you would like to play from a drop down list, along with the game version and client of your choice.
It worked for me, what I found was MY GAME CENTRE ACCOUNT HAD THE SAME EMAIL ADDRESS LINKED TO IT AS MY JAGEX ACCOUNT THAT I CREATED. Then when I clicked on the import character after logging into Jagex it takes you to the login screen for RuneScape, it throws you off because you think I just logged in or I don’t know the password etc that I used which is why I think you have to recover the account first.
I hope this makes sense and helps. Kind regards, TickleYaBean