I think it depends partly on how much of a big deal it would be for your husband to do the 11pm feed. My husband is almost never in bed before 11 anyway, so it's not that huge a burden for him to do an 11pm feed. However if your husband has to be up at 5 and would usually go to bed at 9.45 then it's a bit more of an ask.
It also depends on how hard work your newborn is during the day. If you had
a baby who would only sleep in your arms or in a moving pram (like mine!) I would say your husband should be doing more as you can't really rest in the day. However if she does sleep and the only reason you can't sleep when she is sleeping because she's too noisy, I'm slightly inclined to think you can't be that tired

(I know you'll hate me for saying that! but really after a few months of my daughter I was so tired I could sleep anywhere!)
And you have a nanny too... I can understand she's been focusing on the 3 year old but if I was your husband I think I'd be expecting you to ask the nanny to change things round a bit, so that you can nap, before asking me to do night feeds. Unless your husband agrees that the nanny should focus on the 3 year old.
sorry not the answer you wanted! When does nursery start?