Interpreted the question differently, oddly didn't see that you were looking for their current status, but wrote this anyways. Oh well.

TLDR; The United States created an economic system to govern European affairs. Part of that system relied on the war debts and reparations to keep Europe "in line" and economically connected. It's intent was also to ensure a stable security environment, which Europeans (especially France) would buy into after the First World War. It put heavy pressure on European countries, constraining sovereignty and financial leeway. The war debts were rarely repaid, but left felt and real political restraints. When the system collapsed in 1929-1931, and countries a) faced economic crisis and b) knew they could not be guaranteed by US security as it was clear they were disengaged from European affairs, countries opted out of the 'system,' and therefore they opted out of paying the war debts.

For this answer I will be relying on Adam Tooze's The Deluge: The Great War, America and the Remaking of Global Order, 1916-1931. I know its risky relying on a single source, but Tooze has a particular perspective that I want to use to explain the war debts situation after the First World War.

To better understand the question of debt after the First World War, we need to consider the "web of debts" that bound the major European countries and the United States together.

At the end of the war, the Wilson administration was more interested in creating an economic order that could govern Europe and ensure economic stability, as opposed to a Europe created on self-determination, liberty, etc. Wilson predicated this system on interconnected war debt, reparations, financial stability 'guaranteed' by the United States, and US engagement with Europe to ensure economic stability.

France and Germany were the most devastated by the war, materially and financially, and at the signing of the peace (side note: Tooze argues that France did not force a Carthaginian peace, I go more into this if asked) Germany was to pay large reparations to France. In the years following the war, Germany would often repay France in kind, by sending shipments of coal from the Ruhr mines. From there (and hopefully from payments in German Marks) France would begin to pay back the debt she owed Britain, who would also collect payments from Germany. France and Britain would pay back their enormous debts the United States. Britain pushed for a write-down of inter-Allied claims after the war, and a "clearing of debt," but it was vetoed by the Wilson administration.

As the situation in Europe become more tenuous in 1921, with the inability to repay debts, massive interest accumulation on deferral of payment, a deflationary period, and the newly elected Harding administration with a new "neutral" orientation towards Europe, the United States looked to step away from European entanglement and allow the heavy financial pressure to humble the "passions of the old world." The Harding administration pivoted to a position of disengagement that relied on financial structures within the United States and Europe to influence European direction. Senator Ashhurst of Arizona puts it clearly,

We saved Europe and our Christian civilization. But that does not imply now that the peril is past that we should feed the Europeans and allow them in their great cities to live in idleness and sometimes in luxury.

Another congressional opponent of war debt concessions argued that,

we are taxing the American people as they have never been taxed before...if we collect but the interest upon these loans, we will of necessity reduce the taxation upon our own people by one-seventh.

Bankers in both New York and London disagreed. In a note from Governor Norman of the Bank of England to Benjamin Strong at the New York Federal Reserve,

Having let us suppose, steadied the exchanges by some reparations adjustment, we are immediately to see them unsteadied by inter-allied debt payments.

They blamed Congress for creating a "ridiculous roadblock" between economic stability in Europe. There was a clear divide between American policy and economic and financial realities.

The financial pressure was felt most acutely by France, who relied on German war reparations to be able to pay back the debts they owed Britain and the United States, and as the German economy continued to flounder, the Poincare administration in 1922 became more agitated. Poincare was not opposed to occupying the Ruhr to ensure that the reparations were paid, but knew he could not act on German inability to pay reparations without the backing of the Entente. The French ended up acting anyway, and while they made a substantial return on the occupation, they antagonized Britain and the United States. They also revealed that they could take action into their own hands, a moment of foreign policy sovereignty for France that would set a precedent. The Ruhr crisis also revealed a) how vulnerable Germany was to economic shocks, b) how tenuous the connections were between France, Britain, and the United States, c) how unstable the US system was.

By the mid 1920s, the system became far more complex with the Dawes Plan and other financial agreements, which had the United States making loans to Germany so that it could repay France, who in turn could repay Britain and the United States. A cyclical flow of money that surprisingly kept the system relatively stable. However it relied on US financial institutions continuing to make loans to Germany. Given a situation where the United States would not be economically stable, the stability of the system could be in jeopardy.

This was the prickly situation Europe found itself in. Tied to an economic system of war debts, reparations, financial constraint, and deflation, no country could move against it because of how crippled they were economically, and how much they relied upon the United States to guarantee a system that 'created' stability and security in Europe. A system where shocks and instability would push European countries to more radical action, choosing to move away from the order that had not guaranteed stability rather than towards it. At the end of the decade, Wall Street collapsed and Europe was in turn brought into an unprecedented economic depression. Initially, this tightened the constraints of the US order in Europe. Britain, France, Italy and Germany all chose to stick with deflationary measures as opposed to moving away from the system. This had disastrous consequences. As the depression deepened without end in sight, Britain, France, and Germany looked to freeing themselves from this system that had constrained political and financial sovereignty. In the case of Britain and France, it meant moving off the gold standard and defaulting on debts owed to the United States. After the Second World War, the United States used a different approach to war debts, hoping to avoid another situation like the interwar period.

I realize this was long, but hopefully this gives a clear picture of war debts after the First World War, and why Britain ceased paying the loans in 1931. It's kind of a roundabout way of answering, but I think it gives better context for why Britain stopped paying the loans. I would be interested in hearing other perspectives and criticisms of this interpretation.

EDIT: Formatting. Citation was not fully written out.