EU law definitely applies to your case and if the authorities would decline to apply it, you could go to court to invoke it. In practice, the UK makes it a little more difficult that it needs to with delays and such but it does generally follow EU law. The famous court cases (Surinder Singh, Carpenter…) however are about British citizens using EU law to circumvent the more restrictive rules that apply to them and their family.
In practice, your wife will need to apply for an EEA family permit and then, once in the UK, for a UK residence card. In principle, she has a right to work in the UK without restrictions immediately and should get this visa quickly but in practice I have read reports from people who had to wait several months to actually get the permit they need to come to the UK and work.
One thing I don't know is whether your wife could enter the UK on her other visa and apply for a residence card directly.