A former French prime minister and a previous British head of Nato say that France and the UK have to overcome the risks of Brexit and urgently deepen their military alliance to hedge against the unpredictability of Donald Trump’s White House.

In a report from a taskforce led by Bernard Cazeneuve, France’s prime minister from 2016 to 2017, and George Robertson, a former Labour defence secretary and secretary general of Nato, it is claimed the two countries’ relationship has never been more fragile.

The failure of the UK government to hedge against the dollar ahead of the Brexit referendum, and the subsequent devaluation of sterling, is said to have left a gaping hole in the British defence budget, which has undermined a joint attempt to develop unmanned combat aircraft.

The dispute over the UK’s post-Brexit involvement in Galileo, the EU’s global satellite navigation system, is said to have “emerged as a fault line in the negotiations for the country’s withdrawal from the bloc”, indicative of the breakdown in key joint structures.

At a time when US foreign policy has never been more unpredictable, it is said to be vital to overcome the problems facing the alliance. The alliance was strengthened by a treaty in 2010 through the establishment of the joint expeditionary forces used in Libya but now, it is claimed, it needs updating and upgrading.

The US president, who is likely to be constrained with his domestic agenda following the US midterm elections in which Democrats gained control over the House of Representatives, is expected to turn to foreign policy to make a mark – an area in which he has already built a doubtful reputation.

The report calls for French and British intelligence agencies to work together on a greater scale, for military facilities to be shared, and for concessions to be made in the Brexit talks between the UK and Brussels to bring the two allies closer together.

Lord Robertson: ‘The Trump administration has underscored the need for Europe to do more in its own interests.’ Photograph: Peter Cosgrove/AP

Lord Robertson, who led Nato from 1999 to 2004, said: “I think that the Trump administration has underscored the need for Europe to do more in its own interests and the [midterm election] result doesn’t do anything but underscore the fact that there is an unpredictability about American foreign policy which should drive Europe to do more in its own interests.

“[Trump] has been lukewarm on Nato and has taken a pretty hostile view about the role of some European countries and their seriousness about defence. And that has made European allies more nervous about the ultimate American support.”

Robertson said the proposals were not about building a European army, and he denied the recent suggestion that this was an aim of the French president, Emmanuel Macron. He said it was instead about “reinforcing the European pillar of Nato”.

He added: “This is a crucial alliance between the two major military powers in Europe but it is under pressure due to Brexit and it is has never been more valuable given what America is doing at the moment. It needs to be strengthened, improved and reinforced.”

The report calls on the EU to rethink its relationship with countries that are not member states, to keep the security relationship alive after Brexit, and urges the UK to soften its “red lines” on the jurisdiction of the European court of justice, to allow its security forces access to key databases.

Robertson said: “I don’t think [security] should be part of the bartering to do with trade. There is a clear identifiable interest here in terms of security. Brexit does not change geography.

“Britain and France are close neighbours, close allies and share threats and it is important to say that the two countries account for more than half of European defence spending and probably over 60% of defence capability. We are both in the UN security council, we are both nuclear powers and we have both got the expeditionary capabilities.

“Any EU operation will need UK forces. [The EU’s defence cooperation arm] will only work in operational terms if the UK is involved in it and that is the dilemma we are highlighting.”