Currently purchasing an office building that’s 1900’s era built as a pub originally, I am wanting to turn it into a children’s nursery. through the center of the building is a large square staircase that connects ground, first and second floor, which is open through the center of the staircase, the perimeter is say 3m x 3m with wooden spindles, I believe they are called Atrium staircases, any smoke would funnel straight up the staircase and enter every level.

Would anything need to be done to ensure it’s legal? currently all the doors that open onto the stairway corridor look like fire doors

Firstly you need (it's a legal requirement) a Fire Risk Assessment by a competent specialist who will address this and any other issues. Stairs are, by their very nature, open shafts, but this is not an issue is the stair is a protected route with 30 minute fire resisting construction and 30 minute self closing fire doors with intumescent strips (or depending on a variety of factors possibly even original fire doors with a 25mm rebate).
If the offices were fully legally compliant the stairway should already have this protection as well as a suitable fire detection & warning system and emergency lighting.

This will give you an overview as well as links to detailed guidance: https://www.london-fire.gov.uk/media/7155/gn_03_lf_format.pdf