Pack two small bags rather than one big one.
1 you take in with you for labour. Stock it with LOTS OF CHANGE FOR THE CAR PARK, a towel for you to sit on in the car if leaking waters, a couple of changes of clothes for you (i.e. 2 pairs of pants and 2 nighties/pjs), hair bands, plus something to go home in (leggings, socks and baggy jumper), about 8 maternity pads, and your phone charger, plus a BIN BAG (to put any wet/bloody/pukey clothes in!). In this bag for baby pack anything you need to give the baby its first feeds (nursing bra or formula etc), 5 nappies, a pack of cotton wool/wipes and 2 outfits for baby (short sleeve vest, baby gro, cardigan, hat) plus a muslin and a blanket.
The other bag leave in the car in case you need to be there overnight or for a couple of days. Toiletries for you, a nice towel, hairbrush, LOTS of maternity pads, some more clothes for you, snacks, nice drinks, kindle etc. For baby add in 3-4 more outfits as above plus another blanket and about 4 muslins, the rest of the pack of nappies and wipes/cotton wool, plus more feeding stuff (another nursing bra, nursing pads or more formula etc if you plan to FF). If you have a vaginal delivery they spit up a lot of amniotic fluid the first day or two so you might end up changing them a few times. A small fold up changing mat can be useful.
Baby won't be outside for long and can go home in vest, sleepsuit, cardigan and a blanket or two depending on the temperature outside. Lots of maternity units allow your partner to pull the car outside and so you can pre-warm the car and barely take the baby outside.