UK companies are required to file accounts with Companies House, which has a record of directors, significant shareholders (persons with control), and the accounts. Any changes in control MUST be notified to Companies House.
https://find-and-update.company-information.service.gov.uk/company/00787446/filing-history
Accounts and filing history are publicly available for free.
The accounts filed must meet certain criteria depending on the size of company, and the level of detail they must include depends on the size/criteria that must be met. Mercedes-Benz Grand Prix Limited is a big enough company to be required to file relatively informative accounts.
Currently, Mercedes Benz is majority owned by Daimler UK, which is owned by Daimler AG. Daimler AG at the end of 2020 was the ultimate controlling party due to owning a majority stake. This has not changed.
https://find-and-update.company-information.service.gov.uk/company/00787446/persons-with-significant-control
It is 30% owned by Toto Wolff, who is another person with significant control, during to having >25% (and being a director).
There were press releases in December 2020 announcing Ineos would become a one third equal partner with Daimler AG and Toto Wolff in the team.
THIS HAS NOT YET HAPPENED.
Per the 2020 financial statements on page 6, they note that "upon completion of that transaction", the F1 team would become one third equally owned by each party and that it was scheduled to complete during 2021.
It is 15th December 2021 and that has not yet completed. Ineos are NOT a shareholder in the Mercedes Benz F1 team.
If they had become a shareholder, the company would have been required to file that this had happened, and the persons with significant control would need to be updated (a "person" can be a company in this context).
What does this mean? Likely nothing yet, other than the deal is taking longer to complete than expected, maybe due to Covid, maybe due to other things. But potentially the deal could never happen.
Potentially if Mercedes DID decide to sell the team, they could just change the amount they sell to Ineos.
But currently Daimler AG is still the majority owner of the team, and not in fact an equal one third owner.
EDIT:
Any change in control must be notified within 28 days, so if a sale happens, they would have to update Companies House within 28 days, so things could have completed without Companies House being updated, but only if it was within the last
