I packed at 38.5weeks under pressure from midwife and DH. Did the bag in a few minutes, it's worth to have it ready though. I realised I didn't need half of it, though. I'd recommend to do two bags: one with things you won't need until after the birth, leave it now in the car. One with what you'll need during or before birth.
After: change clothes, babysuits, all-in-ones, baby blanket, make up bag, maternity pads, knickers, brush, towel, dressing gown, etc.
Before / during: snacks, drinks, mobile charger, mp3/ music, tablet with Netflix, change for parking, etc.
The last thing you want to do when in labour is to carry around a suitcase. DH or whoever is with you during labour can be in charge of taking it upstairs. You just take stuff you want with you from the start. That way, if you need to be left at the hospital door whilst DH finds parking (or arrives from work!), you won't have to drag a big bag around.