The certificate fixes every date that follows
The date printed on the certificate of incorporation is not a formality. Every deadline the company will ever have is counted from it, including two you have not heard of yet.
What the certificate actually is
Registering a company costs 100 pounds online and the company is usually on the register within 24 hours. What comes back is a certificate of incorporation, and it carries two things that matter for the rest of the company's life: the company number and the date of formation.
People treat that date as paperwork. It is the origin point of a calendar you did not write and cannot edit.
The four dates it sets on its own
Companies House gives the company an accounting reference date straight away. It is the last day of the month you were registered in, one year later. Register on 11 May and the accounting reference date is 31 May the following year.
From there: the first set of accounts is due 21 months after the date you registered. Every later set is due 9 months after the financial year ends. Corporation Tax is payable 9 months and 1 day after the accounting period ends. The Company Tax Return is due 12 months after the same period ends.
Read that last pair again. The money is due three months before the return that works out how much the money was. Small companies pay first and file second, and nobody explains that anywhere on the way in.
Why the first year does not behave
Because the accounting reference date lands on a month end, a first set of accounts covers more than twelve months. GOV.UK uses 11 May as its own example: the first accounts cover twelve months and three weeks.
A Corporation Tax accounting period cannot be longer than twelve months. So a first set of accounts that covers thirteen months does not fit one tax return, and you file two, with two payment deadlines. One company, one year, two returns.
The part you can control
The only lever here is the day you incorporate, and it is a real lever. A company formed on the 2nd of a month and a company formed on the 28th of the same month share an accounting reference date. The one formed on the 2nd gets almost four more weeks of first financial year for the same deadline.
If the company is not trading yet, that is free time. If it is, it is four weeks of extra bookkeeping in year one.
The dates for your own company come out of the deadline calculator, and the filing itself is described in the dossier.