I'm a photographer (not birth! ) and I'm in a few groups with birth photographers and they are absolutely amazing. They charge a lot because they have to be on call and can be there not just for hours, but days. They can't have other work booked in around their clients' due dates.
I only know of a few in the UK, but it takes a very specific person to make it successful- the ability to be on call, the skill of low-light photography and the creative side.
I got my husband to take a photo of me during labour in the pool with my first and I took a selfie in the back of the car an hour before my 2nd was born. I absolutely treasure those photos as a record of my exoerience but I have never shared them with anyone, so it's often really not about it being the 'insta culture' of sharing every aspect of your life (of course it will be to a few people). It's about documenting one of the most amazing, transformational, emotional, tough, monumental moments of your entire life. Often at a time when your memories will be sketchy through pain and drugs. The beautiful photos that I've seen shared in groups usually show the raw emotions and strength of the woman, the love and support of those around her and the very first seconds of life- its just amazing. A good birth photographer captures all of that with dignity and with barely being noticed.
About your friend- I think it will definitely be hard to get into in the UK, but not impossible. If she's a skilled photographer in the first place and has the flexibility to be on call for clients, then there's no reason she can't give it a go. Like most areas of photography, she'll need to build a portfolio first, which often means working for little money. She'd probably then need to share the hell out of her photos on Instagram and work on her SEO so that the small pool of potential clients could find her. Joining Facebook groups for home births would be my first idea.
I think the main hurdle she'd some across would be the cost. It's not going to viable for her to charge less than £1000 as an absolute minimum. But for most people this is just not something they're willing to pay. Yes, they'd pay it for a wedding photographer where they can share the photos and hang them on the wall. But it's too much for most new parents. Unlike most of the people above, I absolutely would have had a birth photographer, except for the cost- I just wouldn't be able to justify it.
Of the photographers I know about, they often do newborn photos as well, but have to have very understanding clients and clauses in their contracts because they may just have to clear their bookings at a few hours notice.
I'll see if I can find any examples of the photos I love.