All hospitals have different rules when it comes to partner/other family member staying overnight after a baby is born. From personal experience and research, I'm looking to change this and make it become an option whether a member of family can stay or not and to be discussed in your birthing plan.
You may be interested in these active petitions
MY STORY: Upon exiting theatre (around 8pm) after a complicated birth, my husband was told to finish his cup of tea as he needed to leave so I could go up to the ward. This gave us less than an hour as a new family. I had hardly any support from midwives so needed my husband there. This would not put a strain on the NHS as chairs that are already at bedside would be used. Rules that already apply in the day like consuming own food and using other facilities away from ward would still apply.