The majority of the radar, including the AESA array, is from the British (in Edinburgh) part of Selex. SAAB-Ericsson is contributing some back-end parts & software, IIRC. With some lesser bits & pieces, the total British content is about 20%.
The Skyward-G IRST is also from Selex, but I'm not sure whether it's British or Italian.
It's not about company ownership, but where the products are designed & made. Selex is an Italian company, but the British government controls what it exports from the UK, just as the US government controls what BAe Systems or Rolls-Royce or MBDA North America export from the USA. And British controls apply to re-export after fitting in a foreign product, exactly the same as US controls. AFAIK the British government is generally less restrictive, but it wouldn't allow exports of the ES-05 Raven radar to Argentina unless the politics of the Falklands completely changed.