Should I evict a tenant with mental health issues refusing an electrical inspection?

This tenant is exposing your mother to a prosecution (failure to have current electrical inspection) to which there is no real defence (is there a valid certificate? No. Guilty. But but....no buts).
Unfortunately this tenant and their problems will have to become the councils' responsibility. It is possible that once Section 21 is served, the council may realise exactly how they are shooting themselves in the foot and start helping, but i wouldn't bet on it.
Until that happens, your mother has to go the eviction route - remember all costs can be claimed from the tenant.

https://tenor.com/en-GB/view/chinny-reckon-chinny-reckon-jimmy-hill-gif-20013567

The same council will then tell tenant to stay until evicted then give advice were to get legal help to prevent eviction.

I had similar. Report to adult mental health services and flag up the behaviour experienced to date. Give them a clear time line of events of how you are trying to carry out your legal duty. Explain she is now putting herself in direct danger and this is now an emergency situation.

Copy in the council enviro health Dept, head of Adult Social services Dept, and local councillor who heads up the MH committees.

Hit hard - they will all be liable now if anything happens because they have been formally notified of the issue.

𝘈 𝘭𝘢𝘯𝘥𝘭𝘰𝘳𝘥 𝘪𝘴 𝘯𝘰𝘵 𝘪𝘯 𝘣𝘳𝘦𝘢𝘤𝘩 𝘰𝘧 𝘵𝘩𝘦𝘪𝘳 𝘥𝘶𝘵𝘺 𝘵𝘰 𝘤𝘰𝘮𝘱𝘭𝘺 𝘸𝘪𝘵𝘩 𝘢 𝘳𝘦𝘮𝘦𝘥𝘪𝘢𝘭 𝘯𝘰𝘵𝘪𝘤𝘦, 𝘪𝘧 𝘵𝘩𝘦 𝘭𝘢𝘯𝘥𝘭𝘰𝘳𝘥 𝘤𝘢𝘯 𝘴𝘩𝘰𝘸 𝘵𝘩𝘦𝘺 𝘩𝘢𝘷𝘦 𝘵𝘢𝘬𝘦𝘯 𝘢𝘭𝘭 𝘳𝘦𝘢𝘴𝘰𝘯𝘢𝘣𝘭𝘦 𝘴𝘵𝘦𝘱𝘴 𝘵𝘰 𝘤𝘰𝘮𝘱𝘭𝘺.

𝘈 𝘭𝘢𝘯𝘥𝘭𝘰𝘳𝘥 𝘤𝘰𝘶𝘭𝘥 𝘴𝘩𝘰𝘸 𝘳𝘦𝘢𝘴𝘰𝘯𝘢𝘣𝘭𝘦 𝘴𝘵𝘦𝘱𝘴 𝘣𝘺 𝘬𝘦𝘦𝘱𝘪𝘯𝘨 𝘤𝘰𝘱𝘪𝘦𝘴 𝘰𝘧 𝘢𝘭𝘭 𝘤𝘰𝘮𝘮𝘶𝘯𝘪𝘤𝘢𝘵𝘪𝘰𝘯𝘴 𝘵𝘩𝘦𝘺 𝘩𝘢𝘷𝘦 𝘩𝘢𝘥 𝘸𝘪𝘵𝘩 𝘵𝘩𝘦𝘪𝘳 𝘵𝘦𝘯𝘢𝘯𝘵𝘴 𝘢𝘯𝘥 𝘸𝘪𝘵𝘩 𝘦𝘭𝘦𝘤𝘵𝘳𝘪𝘤𝘪𝘢𝘯𝘴 𝘢𝘴 𝘵𝘩𝘦𝘺 𝘵𝘳𝘪𝘦𝘥 𝘵𝘰 𝘢𝘳𝘳𝘢𝘯𝘨𝘦 𝘵𝘩𝘦 𝘸𝘰𝘳𝘬, 𝘪𝘯𝘤𝘭𝘶𝘥𝘪𝘯𝘨 𝘢𝘯𝘺 𝘳𝘦𝘱𝘭𝘪𝘦𝘴 𝘵𝘩𝘦𝘺 𝘩𝘢𝘷𝘦 𝘩𝘢𝘥. 𝘓𝘢𝘯𝘥𝘭𝘰𝘳𝘥𝘴 𝘮𝘢𝘺 𝘢𝘭𝘴𝘰 𝘸𝘢𝘯𝘵 𝘵𝘰 𝘱𝘳𝘰𝘷𝘪𝘥𝘦 𝘰𝘵𝘩𝘦𝘳 𝘦𝘷𝘪𝘥𝘦𝘯𝘤𝘦 𝘵𝘩𝘦𝘺 𝘩𝘢𝘷𝘦 𝘵𝘩𝘢𝘵 𝘵𝘩𝘦 𝘪𝘯𝘴𝘵𝘢𝘭𝘭𝘢𝘵𝘪𝘰𝘯 𝘪𝘴 𝘪𝘯 𝘢 𝘨𝘰𝘰𝘥 𝘤𝘰𝘯𝘥𝘪𝘵𝘪𝘰𝘯 𝘸𝘩𝘪𝘭𝘦 𝘵𝘩𝘦𝘺 𝘢𝘵𝘵𝘦𝘮𝘱𝘵 𝘵𝘰 𝘢𝘳𝘳𝘢𝘯𝘨𝘦 𝘸𝘰𝘳𝘬𝘴. 𝘛𝘩𝘪𝘴 𝘤𝘰𝘶𝘭𝘥 𝘪𝘯𝘤𝘭𝘶𝘥𝘦 𝘵𝘩𝘦 𝘴𝘦𝘳𝘷𝘪𝘤𝘪𝘯𝘨 𝘳𝘦𝘤𝘰𝘳𝘥 𝘢𝘯𝘥 𝘱𝘳𝘦𝘷𝘪𝘰𝘶𝘴 𝘴𝘢𝘧𝘦𝘵𝘺 𝘳𝘦𝘱𝘰𝘳𝘵𝘴.

https://www.gov.uk/government/publications/electrical-safety-standards-in-the-private-rented-sector-guidance-for-landlords-tenants-and-local-authorities/guide-for-landlords-electrical-safety-standards-in-the-private-rented-sector

Does your tenant have a next of kin who perhaps could be approached or perhaps a close friend?

A combination of mental health and strangers in your home can be daunting for anyone and perhaps there is a family member who can attend at the same time as the contractor?

If you consider that the electrical installation is okay, then I agree with other comments but if there could be call for concern, then it needs to be done sooner rather than later.

Adult social care from your local council may perhaps be able to help but they generally cannot discuss individuals in any detail because of GDPR. They may have a team who assist in these situations or perhaps a day care centre where the tenant could be encouraged to visit?

Good luck

Contact Social Services in your mothers name and in writing and state your concerns that your tenant has become a vulnerable person and, as exhibiting clear signs of deteriorating mental health and possibly capacity, you are worried for her welfare. Point out that you’re elderly yourself and need their assistance to organise necessary and required certification.
Copy this to the Housing Department, Benefits Department, Leader of the Council (and Police as already been included in prior communication)
If you do not want to evict, and good on you for not wanting to evict an already vulnerable and likely distressed tenant, you can apply to the Court for an Injunction to gain entry, which could be classed as requiring emergency entry. Then give the tenant 24 hours written notice and I would accompany your tradesman. Does the tenant know you as well as your mother? Would your mother be prepared to go with you? A familiar face may help the situation.
GDPR is causing all sorts of unintended problems and consequences.

I wouldn't just yet. I would contact your nearest carers centre. https://www.nhs.uk/conditions/social-care-and-support-guide/support-and-benefits-for-carers/carer-assessments/

They could be afraid to let people in and may need support and have someone there with them when the electrician goes. Some people who have mental health issues or disabilities wont answer the door to people they have never met even if you tell them someone is coming and what for. It might be because the person coming to do the inspection is a stranger and they may not be able to articulate how they feel about a stranger coming in what is their home, even though you own it. If the mental health issues are bad they may already have some one that visits them or contacts them to support them and they may be known to the local council. So its worth getting in touch with the council first to see if they have a support worker.

Agree with other contributions -- you want to speak to the social worker if there is one, and be sure she gets one if not. If she needs supported housing that's not going to happen without Social Services making the case. And eviction without the right support makes a bad situation worse for a vulnerable person.

This is only going to get worse. I'd serve notice now.