Having the Tories win a narrow majority in 1929 isn't difficult - in OTL, it was a hung parliament with a narrow Labour lead. So basically have something like:
Tories - 300
Labour - 250
Liberals - 50
This means that the Tories are the ones left holding the baby, with no margin for error. That gets you through to 1931, perhaps due to a vote of no-confidence or something.
The problem there is twofold:
- Neither of the two wings of Labour had any clue about fixing the Depression at this point. The Snowden wing thought that Free Trade, Balanced Budget, and Gold Standard was the only way. The Socialist wing thought that they just needed to wait for the final death-throws of capitalism. The only party with anything like a real plan was Lloyd George's Liberals, and they aren't winning 1931 - so if you do have a post-1931 Labour Government, it doesn't end well.
- The Tories were less wedded to the Gold Standard than Snowden. This means that the Depression is going to be less nasty generally.