Please ask your midwife at your next appointment what your hospital's policy is! You might be surprised!
In my first hospital, they asked how we were getting home. I said my mother was picking me up, and the car seat was fixed in her car. No problem, the midwife helped me wheel baby down to the front door in the little magic rocking aquarium cot (people who have given birth will know what I mean!) and helped me get the baby into the car.
Second hospital... oh boy... We were discharged and told we could leave when we were ready. Took a few minutes to pack up, and then husband carried the stuff and I carried the baby in my arms as we walked out of the ward. Said goodbye to the lovely delivery midwife as we passed the office by the door. "Wait, you can't carry your baby. Where's your car seat?" "In the car...?" Turns out their policy, which no one had mentioned, was that I wasn't allowed to carry the baby in my arms in case I collapsed, husband wasn't allowed to carry the baby in his arms in case he dropped her, magic rocking aquarium cot wasn't allowed to leave the ward. Car seat had to be brought to the baby. Well, this wasn't going to be possible as it was a big multi-age fixed-in one. After the midwife had got over the shock of our cavalier attitude to newborn safety, she had to call a colleague to cover her on the ward while she carried our baby in her arms down to the front entrance where my mother was waiting in her car.
I mean, obviously it wasn't a massive problem and we figured out a mutually acceptable solution without too much ado. But I felt like we were the first people she had ever encountered who were mad enough to think they could just carry their own baby .
I'm pregnant with #3 now and giving birth at the second hospital again. Given that we'll take a taxi in, I'm not taking the buggy with me in a taxi and having it sat in various corners of the hospital taking up space. So we've got to decide whether to wing it again like the badass free spirits that we are, or whether to bring a stunt car seat just to carry it downstairs and put it in the real car seat. But we've lost the newborn insert from the bucket seat that came with our travel system, so if they notice we might get in even more trouble, even though the real car seat would be waiting downstairs... And we don't have a newborn-suitable pram any more and don't want to buy one just for my mother to drive to the hospital!
I'm making it sound like a massive deal and it wasn't really. But no one had ever mentioned this "thou shalt not carry thy own baby!" rule to me before that moment so we (midwife and us) were all a bit flummoxed. If you've got room in the car and a suitable pram, chuck it in the boot for insurance. You might be feeling a bit doddery anyway, and it'll save your husband making two trips.