From Kel Richards on the ABC Classic FM Word of the Day:

The sort of expression that you’re likely to hear is something along the lines of: “everything has gone all pear-shaped”. The meaning of this odd little phrase is clear enough – it means that something has gone wrong, that things are not as planned or intended – but where did it come from? It is, I think, still relatively new – at least as a popular, widely known expression. It has largely been put into circulation by the popular ABC police series The Bill – and that indicates that it began life as London East End (or cockney) slang.

Presumably the writers of the series did their research, discovered this colourful phrase, and built it into the script. However, more than that it’s not really possible to say. The origin of this phrase “pear shaped” is not listed in any of the books I’ve consulted. It’s not even in Jonathan Green’s excellent (and exhaustive) Dictionary of Slang.

However, I am prepared to offer the following speculation: that it comes from the notion of a circle gone wrong. A circle is a perfect shape, anything which is clearly not perfect is “a circle gone wrong” – or, in other words, “gone all pear shaped”. There have been other suggestions (including the shape that some women become as they get older), but I think my suggestion is most likely.

I don't have any definitive citation, but James Briggs from Sheffield
Hallam University posted the following:

To go pear shaped is an expression used to indicate that a scheme has not
been perfectly executed. The phrase seems to have originated in British
English in the late 1940s or early 1950s. 1 have come across several
suggested origins, but the best, for me, is related to training aircraft
pilots. At some stage they are encouraged to try to fly loops - very
difficult to make perfectly circular; often the trainee pilot's loops
would go pear shaped.

(<http://phrases.shu.ac.uk/bulletin_board/4/messages/1242.html>)

The same origin is referred to in
<http://www.quinion.com/words/qa/qa-pea2.htm>.

That such a phrase is so popular, without any definitive origin, suggests
that it holds some resonance in its own right. I think the evocative idea
of a pure circle sagging out of shape to become pear-shaped is sufficient to
explain the popularity of the phrase, if not the origin.

Regards,

Chris Veness