I read this post about the idea of making the budget statement the publication of a green paper (a proposal for consultation), and it has quite a lot of merit. Bringing budgets into line with normal policy development would be a welcome change and allow for more influence, not just from the public but also parliament.
The Chancellor’s Autumn statement could be the backdrop for a green paper on the budget. That consultation could last three months until early February, when a draft bill is published with a white paper (describing government policy). Straight away, either the Treasury Select Committee or an elected ad-hoc one could undertake a short pre-legislative look and recommend some changes by early April, as would the Lords’ Economic Affairs sub-committee in a less formal way. The government then would publish its final bill alongside the budget statement. The finance bill would then go through the normal parliamentary processes as described for it here.
It’s not perfect, and you’re squashing the consultation and pre-legislative stuff a bit. Of course, in an ideal world we would have reformed Public Bill Committees and have a thorough scrutiny of the estimates.