I think the notes to the form are written on the assumption that, where the home is jointly owned, the joint owners are the applicant and their spouse/partner. That's by far the commonest situation, but it's not your situation.
Under the Nursing Home Support Scheme Act 2009, either member of the couple can agree to a charge being granted over property to secure Ancillary State Support, and the charge will affect the interest of both members of the couple in the property. The HSE would like the co-owners to sign to indicate agreement, since it means they know what is happening and they agree to it and they are less likely to kick up about it when the time comes to repay the loan. But it's not essential; the signature of one member of the couple is effective to grant a charge over the entire property.
That doesn't apply in your situation. The HSE won't get a good charge over the entire property unless you and your mother both sign. I would think that, if you don't sign, the likely outcome is that your mother's application for ancillary state support will be rejected.