I have overpaid tax in the UK this tax year, and for cash flow reasons, it would be best to get this back as soon as I can by filing a tax return as soon as possible after the year ends. How quickly does the HMRC accept self-assessment tax returns after the end of the tax year?
Checking my records, I have successfully submitted a return on 10th April for the just-finished tax year, and I received the refund (several thousand pounds) on 19th April.
As outlined in this question, you don't need a P60 before you submit if you can infer the figures on it from your final payslip. Likewise you should be able to calculate savings interest yourself from bank statements. If you expect a P11D then you have more of a problem as those are typically not issued for a few months.
HMRC will typically immediately adjust your tax code for future tax years based on the information in a tax return. For example if you claim a certain amount of gift aid, personal pension relief, savings interest and employment expenses, they will assume the next tax year will have the same figures and will issue a tax code designed to get PAYE exactly right the next year.
This may or may not work to your advantage, for example if you make a large one-off pension contribution in year N, they will assume it will happen in future years too and will end up underpaying tax via PAYE for year N+1 and year N+2 until you submit the tax return for year N+1, so you'll end up owing them money - not a problem if you plan for it as it just makes your cashflow better, but you have to be aware of that. In extreme cases if you have a lot of underpaid tax in year N+1/N+2 might even lead to them demanding a payment on account for the following tax year.