No, both the US and the EU systems will provide unencrypted signals which any GPS system can pick up and use, car sat navs etc will all be fine.
The higher end stuff, for the military, and government use will be encrypted and as we will be out of the EU, we won't be given the encryption keys, so the government has decided to spend £100 million just to see how much it would cost to build our own system, which is likely to be between £4-6 billion (and the government has such a brilliant track record of coming in on budget...)
Brexit the gift which keeps on giving...