Does a shareholders' agreement override a company's articles?

No. Section 21 of the Companies Act 2006 provides that a company may amend its articles by special resolution, so an agreement inconsistent with the articles does not change them. Section 33 makes the constitution bind the company and its members as if there were covenants to observe it, which means both documents operate at once and the drafting has to reconcile them rather than assume the agreement wins.

Can shareholders agree that a director cannot be removed?

Not by an agreement with the company. Section 168(1) of the Companies Act 2006 allows a company by ordinary resolution at a meeting to remove a director before the expiration of his period of office, notwithstanding anything in any agreement between it and him. Section 168(2) requires special notice of the resolution, and section 168(5) preserves any claim the director has for compensation or damages for termination of the appointment.

Is a UK shareholders' agreement filed at Companies House?

Usually not, which is the main reason it is used alongside the articles rather than instead of them. But section 29 of the Companies Act 2006 catches resolutions and agreements agreed to by all the members that would otherwise have required a special resolution, as well as class resolutions, and section 30 requires a copy to be forwarded to the registrar within 15 days after it is passed or made, with fines for the company and its officers on default.

Shareholder Agreement in the UK

A UK shareholders' agreement sits alongside the articles, which bind the company and its members as a statutory contract under section 33 of the Companies Act 2006 and are filed at Companies House. The agreement stays private and binds shareholders personally, but it cannot displace statutory rights: section 168 allows removal of a director by ordinary resolution notwithstanding any agreement, and section 994 preserves the unfair prejudice petition.

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Governing law

The Companies Act 2006 governs the constitution of a UK company and the statutory rights of its members, and it sets the limits within which a shareholders' agreement can operate.

Companies Act 2006, s.33

What a shareholder agreement has to contain in the UK

The provisions of a company's constitution bind the company and its members to the same extent as if there were covenants on the part of the company and of each member to observe those provisions, and money payable by a member to the company under its constitution is a debt due from him to the company.
Companies Act 2006, s.33
A company may amend its articles by special resolution, so a shareholders' agreement that contradicts the articles does not change them and both documents continue to operate.
Companies Act 2006, s.21
A company may by ordinary resolution at a meeting remove a director before the expiration of his period of office, notwithstanding anything in any agreement between it and him. Special notice is required, and removal does not deprive the director of compensation or damages for termination.
Companies Act 2006, s.168
A member may petition the court on the ground that the company's affairs are being or have been conducted in a manner that is unfairly prejudicial to the interests of members generally or of some part of its members, or that an actual or proposed act or omission of the company is or would be so prejudicial.
Companies Act 2006, s.994

Form, notarisation and registration

Section 29 catches special resolutions, resolutions or agreements agreed to by all members that would otherwise have required a special resolution, and class resolutions. A copy must be forwarded to the registrar within 15 days after it is passed or made.
Companies Act 2006, ss.29 and 30

What catches drafters out

Trying to entrench a founder's board seat by contract. Section 168(1) allows removal by ordinary resolution notwithstanding anything in any agreement between the company and the director, so the protection has to be built into the constitution and share rights rather than into the agreement alone.
Companies Act 2006, s.168
Assuming a shareholders' agreement never reaches the public register. An agreement agreed to by all the members that would otherwise have needed a special resolution is caught by section 29 and must be filed under section 30 within 15 days, which defeats the confidentiality the parties were relying on.
Companies Act 2006, s.30
Using the agreement to amend the articles indirectly. Only a special resolution under section 21 amends the articles, so where the two documents conflict the articles continue to bind the company and its members under section 33.
Companies Act 2006, s.21