It's certainly not based on any documented incident, and it's a pretty fantastically unlikely story — though it makes for a perfectly enjoyable film. Even if the stated policy of British intelligence was summary execution of suspected German agents (it wasn't), the idea that they would order a clearly compromised and conflicted officer to conduct the execution as some kind of macabre loyalty test falls just this side of ludicrous.

The screenwriter, Stephen Knight, claims to have heard the yarn from the sister of the unfortunate SOE officer in question — though he also admits that "this was a story told to me by an individual whose honesty I cannot verify. Also, in the years since, I have tried many times to find any reference to the incident in the many books which have been written about the SOE. and I have always drawn a blank. "

The biggest problem with that plot twist is that it runs contrary to the primary feature of the British approach to wartime counterintelligence: co-opting, rather than dismantling German agent networks in Britain. From the earliest days of the war, MI5 was strikingly successful in its efforts to interdict German agents immediately or shortly after their arrival in the UK — a task made easier by the fact that the Abwehr was consistently sending low-quality and poorly-prepared agents into the field.

But rather than simply locking up or hanging German agents, MI5 interrogators offered them a choice: either walk free and feed your handlers the information we give you, or hang as a spy. Unsurprisingly, most chose to become double agents; by the spring of 1942, the entire German human intelligence network (around 120 agents) in mainland Britain was being run by MI5 — and successfully fooling the Abwehr into believing that they were receiving a valuable stream of genuine intelligence from the UK. Both facts were independently corroborated by the ULTRA programme's decryption of German signal traffic.

The 'Double-Cross System' (named after the Twenty/XX Committee that managed the programme*) quickly evolved from a run-of-the-mill Security Service counterintelligence operation into a cross-departmental partnership between MI5, MI6, the Government Code & Cypher School and the intelligence branches of all three armed services, and became a key tool for strategic disinformation. Double-Cross agents, for example, were critical to the success of Operation Bodyguard, the vast deception operation designed to convince German military planners that the Allied invasion of Europe would take place literally anywhere but the beaches of Normandy.

So, in the scenario Allied lays out, in which British counterintelligence become aware of a uniquely well-placed German agent with access to the British intelligence services, their likely approach would certainly not be to order her execution — it would be to turn her betrayal to their advantage. The fact that she was married to an intelligence officer, with whom she had a young child, would likely be seen as a bonus that would make it easier to 'double' her.

Some good reads on wartime intelligence and the Double-Cross System:

• Richard Aldrich, GCHQ — the definitive history of British signals intelligence and codebreaking
• Christopher Andrew, The Defence of the Realm — the official history of MI5, written by the best intelligence historian working today
• Keith Jeffery, The Secret History of MI6 — the official history of the Secret Intelligence Service, up to 1949
• J. C. Masterman, The Double-Cross System — written by the Oxford historian who was chairman of the XX Committee
• Stephan Talty, Agent Garbo — an account of the life of Juan Pujol García, the greatest of the Double-Cross agents

Edit: got myself muddled; as /u/plan9plumber points out, the XX Committee was of course named after the Double-Cross System, not vice versa.