First the general principle. If you don't agree with this then you can ignore the rest of the answer:
It is not appropriate to make a professional recommendation for your partner without disclosing the fact of your relationship.
Even if your assessment of their abilities is objective and unbiased, what matters is that you have a very strong potential for a conflict of interest. You have a responsibility as an employee to give your best advice (or to keep quiet), and surely have done so. But the person (or, in this case, organisation) receiving your recommendation should still be permitted to take this conflict into account when deciding to what extent they can or should rely on that advice.
The reason for disclosing conflict of interest is to take out of your hands the assessment as to whether or not you've been objective. And, most importantly, to take it out of the hands of those not able to make objective assessments where they're conflicted. Even if you do happen to be objective: firstly your employer might disagree and prefer to set your recommendation aside in favour of whatever assessment they'd do of a candidate without a recommendation; and secondly even if they do agree it doesn't "look good" to have a company policy that people should be involved in hiring or assessing their own partners. Eventually that policy will trip over someone who isn't objective, which is bad because you hire a duffer and doubly bad because you should have known better.
As against that, your partner has a right in many jurisdictions, as an applicant, not to be discriminated against by virtue of their relationship status. That being the case, they may not want it disclosed at interview that they're in a relationship at all, let alone to you.
My view is that if they wanted not to disclose this then they should not have used your recommendation in the first place, but that's not a qualified legal opinion. It's the one legal concern I can think of, though, that weighs against the basic principle.
The person who handled their application in HR is someone I know well and so I told her that they are my partner in life
Ordinarily, I would say to ask this person. It's HR's job to manage the company's application processes and to make sure that everyone who needs to know about it, knows about it. It's their decision whether the interviewer should know (a) that you personally recommended this applicant; (b) your relationship to the applicant. If you're worried that they've dropped the ball and failed to tell the interviewer something important, check with them. Maybe HR deliberately intends the technical interview to be done by someone ignorant of (a) and/or (b).
So, if HR is managing things normally, then you should not tell your department anything, because by doing so you might upset the company's intended process. But you should confirm with the person you did tell, that HR has taken it into account properly (so for example, if that person thought you were just making conversation and that you'd already made the proper disclosures in the paperwork, they might at that point rush out of the room and do said paperwork).
However, this strikes me as super-weird:
HR told me to also send [their résumé and covering letter] to the technical department they would work for should they get the position, and that department is mine. So I also sent their application to my manager's boss, i.e. my second-level manager.
Normally HR would forward an application to the relevant department, they would not ask either the applicant or the recommender to resubmit the application. So it's possible HR is basically not in the loop of this recruitment process.
If the recommendation and application are just between the department and your partner, with HR taking no role, then telling HR counts for nothing. Then, if you agree with my ethical assessment and you gave your recommendation to the department, you need to disclose your relationship to the department as well as just to HR. Fortunately this is fairly easy: "I've already mentioned this to HR, but just in case they haven't passed it on, I think you're entitled to know when you consider my recommendation that I'm actually recommending my own partner".
If HR is out of the picture, and your department has had the application but never saw your recommendation then I'm not clear. I think it's a bit odd for your partner to be applying to your department and not make the connection known, but I don't think it's in any way deceptive. If you haven't made a recommendation, just passed on an application, then you have not contributed in any way towards the hiring decision, so there's no conflict of interest to worry about.
However it might create a problem: they decide to hire, they assign your partner to work directly under your supervision (or vice-versa), they discover the relationship and realise it's against company policy, they have to re-assign one of you to another role. Pain which would have been avoided if they'd known prior to making the plan. There's a difference between being legally and ethically entitled to cause a headache, and wanting to cause a headache.