Many countries have deep states but not Israel, it seems

The EU is set to exclude the US, UK, and Turkish arms companies from its €150 billion defence fund if their respective countries don't sign agreements with Brussels. That is of course. 

Why did the UK not threaten to leave Nato as a bargaining chip during Brexit negotiations?

Because the UK is ruled by what Dominic Cummings calls the Blob and Messrs Trump and Vance consider is the transatlantic deep state (think William Hague).

Many countries have deep states. Romania (cancelled election) and the UK (Brexit struggle) are two obvious examples.

Some do not, such as Israel. Netanyahu is able to swat away, so far, the opposition of the courts, the generals and the secular Ashkenazi left-of-centre establishment that ruled the country after 1948 until Begin and the Likud (successors to the Irgun) took power.