When a new limited company is formed, Companies House do indeed inform HMRC and in turn HMRC will write to the new limited company at their registered office with form CT41G.
This form provides the Unique Tax Reference for the Company and also allows you or the client to detail if the company will be dormant or trading. In addition, you also inform HRMC of the proposed start date and first yar end for the company.
The three month rule you mention I believe is for Individual/Partnership Self Assessment and not for limited companies.