A few good points raised here that might help reassure your partner:
www.pinsentmasons.com/out-law/guides/coronavirus-can-uk-employers-require-staff-vaccinated
From the employer's, point of view, they would be at risk of being liable should your partner be unlucky enough to suffer an adverse reaction and probably won't want to take that risk:
" Mandating the vaccine could give rise to claims from employees who suffer an adverse reaction to the vaccine if a link can be established, so medical advice for employees may be required."
There is also a data protection issue for employers:
"Requiring evidence of vaccination gives rise to significant data protection issues. Employers would have to carefully consider why they need evidence of vaccination and whether it is appropriate for their business. Doing so will require a data protection impact assessment which must consider not only the reasons for requiring the data but also issues like how it will be held securely, who will have access and whether it is appropriate to hold more than a simple 'yes' or 'no'."
Also, the employer should consider why they are mandating it or threatening dismissal and if there are better alternatives:
"One of the challenges for employers will be in justifying why accommodation can be made for those who are, for example, pregnant, and not for other employees. This is likely to go to the proportionality of the approach and will be relevant should there be any legal challenge to a dismissal.
An employer seriously considering dismissing an employee as a result of a refusal to be vaccinated will need to give careful thought to whether there are any alternatives to dismissal - for example, reallocating the employee to another role where this does not amount to a detriment in the particular circumstances. Legal advice should be sought before taking any action that might involve any detriment, including dismissal, to an employee who refuses vaccination, given the fact-specific nature of these issues."
It sounds like employers should seek legal advice too!