I had exactly this situation. If I had known then what I know now, I would have just taken the sling in and sat belted in, rear facing in a black cab.
However, I must alert you to what happened to me when my LO was born 2 months ago, so it doesn't happen to your daughter - the hospital (St Thomas on Westminster Bridge), told us that they wouldn't let us leave unless we had a car seat, but for those of us who didn't have cars, the local minicab company (for which the hospital provided details) could transport newborns and provide a car seat. We rang the cab company and checked, and they confirmed this was the case.
On the day, we booked a cab, telling them we needed a cab with a car seat for a newborn and the car that turned up arrived with an uninstalled seat that was totally unsuitable for a newborn - it was a seat for a much older child, and would have been unsafe. The cab driver had no idea how to install it and when he tried, we saw from the picture on the seat that he was installing it upside down. We called the cab company back and they told us that this was the only seat it had.
As we didn't have the sling with us, and I had just had a difficult labour and couldn't walk more than a few metres at a time, we wound up calling a friend of DH (who was the only person we knew with a baby and a car) and begging him to do us a huge favour and come and pick us up.
So, if the hospital recommends a cab company who can take you home, don't take their word for it, do a trial run before the big day.
But I would recommend taking a sling instead if you are too far away to just walk home with the pram, or if your daughter is in no fit state to walk.
And congratulations on your grandson!