I don't think you should try discussing the matter again with the nanny.
She obviously felt that she was being accused either of stealing or of negligence when your husband asked if she was quite sure she had locked the conservatory door.
You discussed the matter with them both, and your husband has apologised.
Nanny refuses to have your key - did she have one before this incident?
I can see both sides of this. In the nanny's place I would be hesitant about holding your key, after this incident.
Your husband was surely justified in asking her if she could possibly have forgotten to lock a door, when he suspected there had been a break-in, but either his query sounded like an accusation, or the nanny was offended without cause.
Either way, least said, soonest mended from now on, I should think.
If you are planning to go away soon, find a neighbour or colleague who is willing to have an extra key to your house and go past to check that everything is all right.
If you do this, do talk to the nanny about it beforehand.
If you live in an area with frequent break-ins, would it not be advisable to install a proper alarm system? You trust the Nanny with a key, which she prefers not to take home with her, so surely you would be willing to trust her with a code to the alarm system. These days, everyone using the system can have an individual code, so there is no doubt about who comes and goes when, or who may have forgotten to set the alarm.