I am somewhat confused by this and may be labouring under a misunderstanding.
Employees are given a staff oyster card and they can nominate someone else to hold a nominee card.
On that basis your partners nominee card is a nominee card they have been nominated to have by someone else (an employee) who is not you.
Your use of the nominee card is against the rules and is treated very seriously by TfL.
You will have to convince TfL that use of the card was an error on your part but this presents a mountain to climb because they will enquire how you were in possession of the card in the first place.
If the card holder gave it to you or you took it with their knowledge then not only are you in trouble but also the card holder and the staff member.
If you took it without the card holders permission then this could be viewed as tantamount to theft - a serious matter indeed.
EDIT just seen the other posts posted whilst I was writing this reply.
Using your partners bank card undermines the whole security of the card and the associated bank account but that is not the issue.
It furthers calls into question your actions and whether your use of the card was known about by the card holder.
I am genuinely not sure what advice we can give other than to apologise most sincerely whilst somehow endeavouring to convince them it was a mistake but as I am sure you realise the opposite construct of events - that you were doing this deliberately to avoid paying the fare - is equally if not more likely the explanation for this event.
When TfL analyse the card usage are they going to see similar journeys (start and finish, time of travel) as the one when you were caught using the nominee card?
If so it becomes even more unlikely that your usage was a mistake.
@Hadders usually gives appropriate guidance and advice and doubtless he will join the conversation sometime soon.