I suspect the UK might lose its nuclear deterrent (and with it, its permanent seat on the UN Security Council) before 2020, thanks to Donald Trump. Here's why.
Background: The UK was the third country to test a nuclear weapon, in October 1952; it's one of the five recognized nuclear armed states under the non-proliferation treaty, and since the 1958 UK-USA Mutual Defense Agreement it has cooperated closely with the USA.
The stated goal of the British strategic nuclear deterrent since the mid-1950s has been to deter a Soviet nuclear strike on the UK (and, rather more murkily since 1991, a Russian attack). As such, the stated mission was to be able to wipe Moscow and surroundings off the map in the event that the UK was attacked and the USA declined to get involved in a strategic nuclear pissing match on behalf of its ally. (Whether or not this scenario makes any sense whatsoever is a moot point. Less controversially, the acknowledged strategic nuclear force is pointed to as an argument for the UK's continuing permanent seat on the United Nations Security Council.) During the 1950s-1980s the Royal Air Force maintained a strategic bomber capability in the shape of the V-Force, while from the mid-1960s onwards the Royal Navy operated Polaris SSBNs, but air-delivered nuclear weapons were phased out from the early 1980s onwards, along with all tactical nuclear weapons.
Since the 1990s, the British nuclear deterrent capability has relied exclusively on the Trident program. The UK built and operates four Vanguard class nuclear-powered ballistic missile submarines (equivalent to but slightly smaller than the corresponding US Ohio Class SSBN), carrying UGM-133 Trident II ballistic missiles leased from a common pool with those of the US Navy, but with warheads built in the UK. As SLBMs can't easily be serviced inside the cramped launch tubes on board a submarine (unlike land-based ICBMs, which have rather more spacious accommodation), they are returned to the Strategic Weapons Facility Atlantic for regular maintenance and replacement; it is believed warheads are removed and reinstalled at HMNB Clyde on Faslane, 25 miles from Glasgow.
As the Vanguard-class submarines entered service in the early 1990s, they've been operating for 20 years already. The Atlantic ocean is a harsh environment, and nuclear submarines aren't immortal; the British government therefore committed in 2016 to procuring four new Dreadnought-class SSBNs, which are intended to carry updated Trident D-5 missiles and to enter service from 2028.
Here are two possible reasons why this won't happen.
Working hypothesis #1: Donald Trump is an agent of influence of Moscow. Less alarmingly: Putin's people have got blackmail material on the current President and this explains his willingness to pursue policies favourable to the Kremlin. Russian foreign policy is no longer ideologically dominated by communism, but focusses on narrow Russian interests as a regional hegemonic power and primary oil and gas exporter.
Clearly, it is not in Russia's geopolitical interest to allow a small, belligerent neighbor to point strategic nuclear missiles at Moscow. But this neighbor's nuclear capability has a single point of failure in the shape of the resupply arrangements under the 1958 UK-USA Agreement. Donald Trump has made no bones about his willingness to renegotiate existing treaties in the USA's favor, and has indicated that he wants to modernise and expand the US strategic nuclear capability. Existing nuclear weapons modernization programs make the first goal pointless (thanks, Obama!) but he might plausibly try to withdraw British access to Trident D-5 in order to justify commissioning four new US Navy SSBNs to carry the same missiles and warheads.
(Yes, this would break the "special relationship" between the USA and the UK for good—but remember, this is Donald Trump we're talking about: the original diplomatic bull in a china shop who decapitated the state department in his first month in office.)
Trump could present this as delivering on his promise to expand the US nuclear capability, while handing his buddy a gift-wrapped geopolitical easter egg.
Working hypothesis #2: Let us suppose that Donald Trump isn't a Russian agent of influence. He might still withdraw, or threaten, British access to Trident as a negotiation lever in search of a better trade deal with the UK, when Theresa May or her successor comes cap-in-hand to Washington DC in the wake of Brexit. It's a clear negative sum game for the British negotiating side—you can have a nuclear deterrent, or a slightly less unpalatable trade deal, but not both.
In this scenario, Trump wouldn't be following any geopolitical agenda; he'd just be using the British Trident renewal program as a handy stick to beat an opponent with, because Trump doesn't understand allies: he only understands supporters and enemies.
As for how fast the British Trident force might go away ...
Missiles don't have an indefinite shelf-life: they need regular servicing and maintenance. By abrogating the 1958 agreement, or banning Royal Navy warships from retrieving or delivering UGM-133s from the common stockpile at King's Bay, POTUS could rely on the currently-loaded missiles becoming unreliable or unsafe to launch within a relatively short period of time—enough for trade negotiations, perhaps, but too short to design and procure even a temporary replacement. It's unlikely that French M51 missiles) could be carried aboard Dreadnought-class SSBNs without major design changes to the submarines, even if they were a politically viable replacement (which, in the wake of Brexit, they might well not be).