The problem is now France - they've just gone through a "Suez" moment and it can be expected to cause all sorts of issues going forward.
Let's say co-operation will be limited - that applies in places like the Sahel and Djibouti. They could decide to get closer to China - or at least less obstructive. That could open up most of the S Pacific to the PLA(N) - running right through any US-Australian "defence line"
I can't see them being very keen to help out the UK with the illegal immigrant problem either
They will be less helpful at the EU, NATO and the UN etc etc
So there will be price to pay