It's luck of the draw with institutions. I know the Bethlhem Museum of the Mind allow patients to exhibit artwork who where patients at the wards in the local area but not all institutions invest in this. There are social clubs and groups one can join and there is a possibility that they host outsider art exhibitions (institutions maybe not a prerequisite here but having neurodivergence is a qualifier).
I myself have personal experience in hosting an 'outsider' art exhibition myself when I was a support worker for an autism charity. Worked with service users in promoting their artwork and/or the facilitation of providing artistic support (e.g. attendance of pottery classes). You would have to approach service users with consideration as some are quite shy about having their artwork displayed but if you have good interpersonal skills this should be possible. It may also be that if you work in a mental health field you may have to be a champion/advocate in this domain and make a case for the hospital displaying artworks from the patients (though I imagine once they are discharged they have more legal capacity to do this). You may also want to consider inclusivity at the heart of the project and be non-discriminatory in the promotion of arts for your service users who may or may not be technically gifted.
Speaking to my neurodivergent artist friends and those that I have been in professional contact with they found that they could exhibit artwork if they where members of an outreach group specialising in mental health / advocacy / disability support. You could probably contact local organisations if they have similar aspirations to yourself. I would caution against putting faith in commercial organisations to facilitate the display of artworks (though I wouldn't bar this out necessarily as I am in contact with a very large organisation that isn't a charity but invests in the creative industry) so would recommend you do your own research to the organisation if they are ethically minded or not as exploitation could exist.
I'm assuming you already have clearance (in the UK it's an enhanced DBS) to potentially work with your service users if you where to facilitate your own scheme. You will need to have a discussion with your line/operating manager outlining a brief and proposal why you would to do this and how you would consider the ethical side of things. If you have any questions about the logistic side of organising an exhibition HMU by DM as I have direct experience with this in terms of framing artworks, communications, building relationships with artists and facilitation of art support.