I used to do my taxes by hand: it was actually faster than the online system I use now - mostly because it was a lot quickly to skip the section dealing with rail road retirees and such. I quit doing it only because one time I missed copying line 13 of form 1234 to line 47b or form 5678 and got the wrong numbers. A simple mistake that the IRS caught right away, but then I had to go refile my state taxes which depended on the wrong numbers: that was painful enough I quit doing it by hand. (why the state didn't also catch the mistake and avoid the whole refile garbage I don't know)
When an employee leaves a job they are given a P45 form to hand to the new employer, which contains a summary of any pay to date for the current tax year along with a tax code.
All I have to do when hiring is enter this information into the accounting software and everything just works. National Insurance, Pensions and Student Loans are calculated monthly and taken via Direct Debit, as are VAT payments and returns. Employees are automatically emailed payslips with breakdowns of their pay and deductions.
Any relevant information is submitted to HMRC (the UK's IRS) by the accounting software, so it takes me ~10 mins a month to complete payroll for 5 employees. When an employee leaves, creating a P45 for the next employer takes a single click.
The gov.uk website is exceptionally well organised and contains detailed lists of everything an employer or employee needs to do at all stages in the process.
The US self-assessment system seems opaque, inefficient and ripe for abuse in comparison. It also places the stress of reporting taxes onto every individual rather than just the employers.
Honestly I can't think of many ways to improve the UK's online accounting process from either an employer or employee's standpoint because I already spend less than a day total per year on it.
The annoying thing is that the U. S. does this, too. They just wait until the numbers don't match before they tell you what their number is. Hey, why don't you tell me that number up front and save us a lot of trouble? Because I just paid several thousands dollars recently because our numbers didn't match up. The IRS already knew, "here's your bill". A lot of back-and-forth could have been saved if the IRS said up front before I filed, "here's what we've got, just to give you a ballpark figure to work with and let you know if we're lining up, or have a problem."
Here's how it could work, based on a real-world example from another country:
Before each tax season begins, the IRS sends each taxpayer an estimate of how much they are expected to earn that year, based on previous years. They are assigned a tax bracket based on this estimate. If they think the estimate is wrong, they send back a new number and their tax bracket is adjusted accordingly. The taxpayer can also give estimates of their deductions at this point, for a more accurate estimate of the tax bracket.
Employees withhold taxes from wages before payment, based on the tax bracket each employee is in. They report the wages paid and pay the IRS the withheld amount.
Investment funds etc. are required to report gains and losses for each of their clients to the IRS.
At the end of the year, the IRS sums up the income they know about from these reports, compare the sum due to the sum already paid from withheld taxes and send the taxpayer a summary. If the taxpayer has no other income to report, and has no complaints about the data the IRS already has, they don't need to file anything further. The report comes with a bill for taxes remaining, or if too much has been paid, a summary of how much will be returned.
For the majority of taxpayers, that is, those whose source of income is wages from employment or government benefits, all this takes maybe 10 minutes a year to handle. Some people still need to file, but they're a minority.