How can the mother of an EU citizen quickly receive a UK visa?

If your mother is dependent on you or your spouse, she is eligible for an EEA family permit to travel to the UK with you. The EEA family permit is free of charge and should be issued more quickly than a standard visitor visa.

If you are traveling by train or ferry, you can also try presenting proof of your relationship and her dependence on you at the border. This follows from The Immigration (European Economic Area) Regulations 2016, at 11(4):

(4) Before an immigration officer refuses admission to the United Kingdom to a person under this regulation because the person does not produce on arrival a document mentioned in paragraph (1) or (2), the immigration officer must provide every reasonable opportunity for the document to be obtained by, or brought to, the person or allow the person to prove by other means that the person is—

(a) ...

(b) a family member of an EEA national with a right to accompany that EEA national or join that EEA national in the United Kingdom;

(c) ...

(emphasis added)

That is obviously a riskier proposition, but if you don't have a couple of weeks for an EEA family permit application, it might be worth trying.

This answer will cease to be valid if the UK leaves the freedom-of-movement scheme, as seems likely.