They can support their partner by a) looking after themselves which is not possible on a crammed ward, and b) giving their partner space to recover when needed.
If a woman isn't coping, the nursing staff need to evaluate their care. No one in hospital should ever be left feeling alone and scared. Nurses are there to care for the whole person, and it's part of our job to make those in our care feel comfortable, settled and reassured.
It's natural to feel anxious at first, and it's bound to be hard caring for a baby straight after surgery or even after a natural delivery. But that doesn't mean we should be herding in their partners to be with them 24/7 just as if one of my patients was frightened, I wouldn't encourage them to call their husband/wife to sit beside them for the night because it just isn't safe.
Wards need to be as uncluttered as possible so that in the event of an emergency - a rapid transfer, an arrest, a security breach, a fire - an evacuation or movement can be carried out rapidly and safely. In an average ward of 36 patients, if you imagine 30 of them have partners your number of people to keep track of is now 66 at any given time. 66 people all needing to pee, eat, shower. 66 lots of luggage and 30 extra chairs for overnight. That's just not safe, feasible or practical at all.
If a woman gives birth to a child that's born very seriously ill, or if the woman herself becomes poorly and needs the care of an HDU etc, that might be handled differently. But for the majority of mums, cared for on multi bed bays, it shouldn't be possible for a partner to hang around 24/7.
Beyond anything else, what can your partner really do to help you? They cannot touch your IV, or drains. They can't check your stitches. They should never be helping you to mobilise (talking from experience , 'help' can actually be very dangerous to the patient). They can't get you more towels, linen changes, more blankets. They can't make you a cuppa. They can't take your observations, make sure you're not suffering from complications. Overnight especially, there's very little they can or would hopefully need to do that a nurse can't.