I have 3, now aged 5, 4, and 2. I never had a nanny - although it sure would have been nice! My husband was able to take time off with each birth, but last time he took just 10 days.
You can do it. My boys are close in age like yours will be and it actually made it easier in a way. My oldest was still little enough to not need to be out and about and busy all the time when his brother was born. He was still entertained easily at home with books, playdough, coloring, little videos etc.
Your oldest must be in school all day, so that will help too.
My advice is give yourself some slack, and don't expect to do too much too soon. Ease into it and relax. Your 16 month old will be fine if she watches 20 minutes of video everyday so you can take a shower... :)
Good luck and congratuations!
J.