Taxis are considered public transport and are usually legally exempt. However, Uber does not fall under the same classification and seem to have a mixed success rate. If at all possible, for your safety and your newborn's, I'd bring a carseat and install it for the ride. The seatbelt method is typically easier.
Legally: no Morally/safety wise: yes. Please do not hold a newborn in a taxi, it's so incredibly dangerous
Nuna urbn car seat is really good because has isofix without a base, so if the car is newer than 2014 you can clip it in more securely than just with a seatbelt (or you can use seatbelt if the car is older)
@Holly isofix is no more safe or secure than a seat belt installation, it's just a convenience feature
I take our car seat to Uber rides as I don't drive.. if I need to run an errand after, i bring the collapsible pram base so I can attach my carseat to the pram after..
They should have same regulations as a taxi in which you don’t need to have a car seat as long as your comfortable holding baby/ toddler x