Rule 1: Going out with a baby always remember that it is better to be overpacked than underpacked. Extra clothes, extra diapers, extra toy, etc.
The longer the outdoor trip, the more you you need.
One thing about Speedy is that the opening is really fix (can’t open wide) and it’s a dark black hole inside. It’s alright for normal use and most occasions.
I would prefer to use a bag that can open widely for a baby bag. (Imagine baby screaming for the toy or the leaking diapers panic). A bag that opens widely is easier to find stuff. So I agree with the recommendations of a Neverfull. However, personally I prefer a bag with a zip because I don’t like thugs dropping out.
So my recommendations are - either get a big bag for baby and you or get a big bag for baby and a mini bag for your personal things like phone and wallet.