It is not currently possible to get a single mumps vaccine in the UK. It is not known whether this situation will change as the mumps vaccines produced other manufacturers are not licensed in the UK.
You could easily get her vaccinated against measles and rubella, but leave her at risk of catching and then transmitting mumps to others who are unable (rather than have chosen not to) to be vaccinated. Mumps is not a disease to be shrugged off. It can have serious complications such as:
Infection of other organ systems
Mumps viral infections in adolescent and adult males carry an up to 30% risk that the testes may become infected (orchitis or epididymitis), which can be quite painful; about half of these infections result in testicular atrophy, and in rare cases sterility can follow.
Spontaneous abortion in about 27% of cases during the first trimester of pregnancy.
Mild forms of meningitis in up to 10% of cases (40% of cases occur without parotid swelling)
Oophoritis (inflammation of ovaries) in about 5% of adolescent and adult females, but fertility is affected in almost half of these 5%.
Pancreatitis in about 4% of cases, manifesting as abdominal pain and vomiting
Encephalitis (very rare, and fatal in about 1% of the cases when it occurs)
Profound (91 dB or more) but rare sensorineural hearing loss, uni- or bilateral. Acute unilateral deafness occurs in about 0.005% of cases.
While these may not concern you unduly as your dd is 12 months old, there will be teenagers and adults who cannot be immunised who mumps poses much more of a risk to.
I'm not sure that I follow your argument of MMR being political rather than medical.
In answer to your question about where you could get your child vaccinated, a simple Google will be sufficient. May I urge you make your decisions and get her immunised as soon as possible as the incidence of mumps peaks in late winter and spring.