First point - the VAT registration threshold doesn't specifically apply to tax years, it applies to any twelve month period on a rolling basis. This is why it should be checked monthly.
You will be required to register once you exceed the limit. So if you exceeded it during March 2022, you would have had 30 days (so until 30 April) to register, and your registration would have been effective from 1 May. You wouldn't have to pay over VAT on any of your 2021/21 turnover, as it would be pre-registration. But you would be liable for VAT on your turnover since 1 May 2022, the date you should have registered.
However, HMRC have the discretion to grant "exception" to the requirement to register for VAT where it can be shown that your turnover for the first year following the potential registration date would likely be less than £83,000 (the limit at which a VAT registered business can de-register). This is designed to prevent businesses having to register just because of one-off transactions. Such applications can be made late, where it can be shown that the information necessary would have been available at the right time (i.e. you would have known last May that your turnover for May 2022 to April 2023 wouldn't exceed the limit). But I expect that your accountant will already be looking into that.
For info, HMRC's public guidance on VAT registration is here: https://www.gov.uk/register-for-vat and their more technical internal guidance (intended for use by HMRC officers) on late requests for exception from registration is here: https://www.gov.uk/hmrc-internal-manuals/vat-registration-manual/vatreg19150