Specifically in Westminster systems (like Canada, Australia, New Zealand, and the UK) the budget is a supply bill, a bill where the government asks for money.

Specifically in Westminster systems (like Canada, Australia, New Zealand, and the UK) the budget is a supply bill, a bill where the government asks for money. Supply bills are automatically confidence motions, which means that if they fail to be passed then the government (i.e, the Prime Minister and their Cabinet) is said to have lost the confidence of the House. If this happens, the Prime Minister loses the right to govern and a new government must be formed. Usually this means an election, but in rare cases the opposition parties can form a coalition that has the confidence of the House. This process is mediated by the Governor General or (in the UK) the Queen. But in all cases, until a new government is formed and they manage to pass a budget, the old budget remains in effect. There's no scenario in a parliamentary democracy where failing to pass a budget will cause the bureaucracy to shut down.