bednobs is right that there's no automatic right for the employer to let a person go part time, but they will have to give the reason why the employee can't do it.
If she hasn't already, your wife should make her application in writing, stating the hours she proposes to work plus how she sees the remaining work being covered e.g. by a job share arrangement or by them employing another part timer.
The employer then needs to respond with, if as seems likely, the reasons why they can't do this - this could be for financial reasons etc.
To a large extent it will come down to the size of the company. With a large company it will be expected that they could fairly easily make adjustments, employ another part time person etc. With a small company this isn't often viable so the employer would have more reason to refuse the request.
If the employer is found to be simply being unreasonable then your wife would have a case for taking them to a tribunal under the sex discrimination laws. It does though depend on the factors above. Get a response from them in writing and see what case they have put up for denying her. It's not cut and dried on either side and the employer would need to be careful.