It's not necessary to have paid contributions in the last complete tax year - the contributions can be either paid or credited. Most people applying for Invalidity Pension would be applying on the basis of their credited contributions given the fact that it is usual to be claiming Illness Benefit for at least 12 months before their claim.
When claiming Illness Benefit your relative will have been receiving credited contributions, so there is no need for them to worry on that score.
My advice would be to wait until they have been in receipt of Illness Benefit for 12 months & then go ahead & apply for Invalidity Pension, although in reality, given the fact that their GP has said that they will probably not be able to work again, they could actually apply now, but there is no need to worry about the contributions.
Either way the process will usually take some time, but the transition is fairly streamlined & there will be no disruption to his/her payments, & your relative should continue to claim Illness Benefit until his Invalidity Pension comes through. They will be informed of when to stop sending in the certs as soon as the application is approved & ready to be paid.