Silverstone is deep in the country, and local taxis in such places (which are just standard 4-door cars) are way cheaper than in London. Adjacent small towns are Brackley and Buckingham. Just put "Taxi Brackley" or whatever into Google and you will see a couple of companies. Phone them up the day before (they're not high-tech website operations) and book the driver to meet you at Heathrow. It will be a common run for them.