What about - yes, it being Thatcher - but some radical Communist/Trade Union guy (that would then be a radical Syndicalist) manages to hit Thatcher during some speech either directly after the Winter of Discontent or in 1983/4 (when Scargill and the Unionists were most active, AFAIK)?
Then, conspiracy theories about support from Moscow and such things will abound...
And I will present a scenario... hopefully not too offensive for English football fans:
POD: March 1979 - in a football game against Preston North End, a certain Gary Lineker is rudely fouled. The defender sees the red card, but Gary Lineker's damage is much worse: A complicated knee injury which ends his professional career as a young talent.
Instead, Lineker at first jobs around and then joins British Rail, and thus gets into contact with the unions, the TUC and other such left-wing organisations. He is radicalised during the Thatcher administration etc.
On September 15, 1983, the first PM since the Napoleonic Wars is assassinated: Margret Thatcher during a speech originally wanting to proclaim an uptick in the economy due to her harsh laws on trade unions and strikes, and the newest results in the Falkland war. Soon after, Gary Lineker is apprehended and arrested as the assassin.
When, on October 9, 1983, Gary Lineker is bludgeoned to death by another inmate, conspiracy theories similar to the Kennedy assassination theories begin to sprout like mushrooms.