There are so many elements to this.
Through my work I come across a lot of people who struggle to access NHS Mental Health services for many reasons.
They are told they can't have therapy until they've stopped drinking to excess or taking drugs, but the reason they're drinking to excess or taking drugs is because they've been self-medicating for low mood, childhood trauma or anxiety. So they're signposted to AA but the religious element and the potential awkwardness/social anxiety of group support is alienating to some people and it also doesn't address the Mental Health reasons why they became dependent in the first place.
I've worked with people who have had to turn down therapy on the NHS because they couldn't take time off work without disclosing the reason to their employer and they felt they would lose their job, be treated differently or not be supported if they spoke about their Mental Health. I've also worked with single parents who couldn't attend appointments due to having no childcare or because they have caring responsibilities for an adult partner or family member and no one to cover.
There might be cultural barriers, stigma or lack of understanding on the part of Mental Health professionals about the persons specific ethnic and cultural background and how this impacts on their experience of emotional distress. There are some communities where it is still widely believed that Mental health problems do not exist, or where talking to anyone outside of your own family about problems is very much discouraged.
Staffing and resource is a massive issue. I worked with a child who was open to the Mental Health crisis team for intensive 'home treatment' as there were no hospital beds available. They were visited by a different MH clinician every time so they never got to trust or build a rapport with any of them. The people I work with who are open to Adult MH services experience frequent change of workers, temp agency workers, workers go on long term sick with stress etc. Having to tell your story over and over, getting to like and trust someone then they disappear and are replaced by another stranger, can be really difficult.
Having to navigate the endless beaurocracy of the system is also pretty soul destroying. Being bounced around between different services and teams, endless form filling and phone calls, then when you finally get to see someone for an assessment after waiting months or even years they tell you they're not the right service for you so you need to go back to your GP and ask for a new referral, to be placed on another waitng list and no one properly explains who does what and what is the difference between all these teams and services...it's enough to drive you mad on its own.
To be honest, I've also worked with people who have had wildly unrealistic expectations of what "help" for their Mental Health looks like, so then they inevitably feel let down by what is actually available. Some people also don't understand that the act of 'going to therapy', as in turning up every week and sitting in the room for 50 minutes, is unlikely to lead to meaningful change on it's own. You generally have to put the strategies into practice outside of your sessions, not just talk about them. Some people don't want to do that, or don't feel able to do that. Some people also seem to pin all their hopes on therapy but then aren't willing, ready or able to make other changes in their life, work or relationships to address the root cause of their issues, so they blame the therapy or the therapist when they don't feel any better.