Congrats on your newborn.
I have a 20 month old and a 4 month old and do manage to get out of the house on time which has surprised me because I am so scatty!
I just deal with the stuff that needs dealing with before I leave which is just to get us all washed, teeth brushed, dressed and fed before we leave
Leave the house mess until later on in the day! Sod the house / kitchen mess you have enough time when you get back to tackle it.
DS1 wakes up at 6/6.30 so I start with putting him on an ottoman for him to brush his teeth and that entertains him for a while ahaha whilst I get ready or get bags ready.
In your case I leave toddler asleep whilst you get yourself ready and the newborn.
Pack baby bag night before if you are pushed for time.. or less stressed morning.
When toddler wakes up just get him ready to go out and fed them leave
With my 4 month old I sort of know when he's gonna nap so I will get him ready first and fed and put a bib on him in case he spits up milk (last thing I want is to change clothes again esp when we have to go somewhere). Once he is fed and dressed I will put him in the baby seat ready to leave (obviously he is supervised at all times whilst in the seat)...if he falls asleep I will put him in baby seat before I get eldest ready at least DS2 is sorted.
If DS2 is not asleep he is still fine in the car seat because he has been fed and knows he is going out.
If it's too much going to toddler groups first thing in morning maybe look at ones that take place at 10/11/12 so you have more time in the morning and you are more relaxed
Good luck