Expectant father Chris Kaba, 24, was shot once in the head through the windscreen of an Audi he was driving in Streatham after a police chase on September 5.
The Independent Office for Police Conduct (IOPC) said it had launched a criminal investigation which will limit what they can release to the public.
In a statement, the IOPC said it was required by law to conduct a “wide-ranging independent investigation”.
The statement went on: “Therefore, we must explore all of the circumstances surrounding Mr Kaba’s death including: how the officers came to be aware of the vehicle Mr Kaba was driving; if they had any prior knowledge of Mr Kaba; and their decision-making and actions on the 5 September.
“We will also examine whether or not Mr Kaba’s race influenced any actions taken by the police.”
The Metropolitan Police firearms officer who took the fatal shot was later suspended from duty.
The IPOC said it had been in contact with Mr Kaba’s family regarding the investigation, which is expected to take six to nine months to complete.
The statement added the family would be “privately and confidentially” given the opportunity to view video footage of the leadup to the shooting “as soon as it is practicable”.
Mr Kaba’s cousin Jefferson Bosela said: “We want the footage to be shown to the family to have a clear understanding of what happened.
“We want a timeline of how long the investigation will take because the IOPC are renowned for having very lengthy investigations. We want it to be swift, we want it to be effective and we want it to be thorough.
“We won’t be waiting years for this. We want justice as soon as we can.”
In a statement last week, his family said: “We are devastated; we need answers and we need accountability.
“We are worried that if Chris had not been black, he would have been arrested on Monday evening and not had his life cut short.”
It has become known that Kaba was expecting a child with Ms Waite, who described him as a “very kind, loving and caring young man”, said South London Press.
Separately, a coroner’s inquest into the death will be opened on October 4.
The Mayor of London, Sadiq Khan, said: “I fully understand the grave concerns and impact of Chris Kaba’s death on Black Londoners across our city and the anger, pain and fear it has caused across our communities – as well as the desire for justice and change.
“The IOPC have confirmed to me that they are fully committed to carrying out a thorough and comprehensive investigation to establish all of the facts – with all key findings made public. I’m clear that the independent investigation must be fearless and leave no stone unturned.”
Assistant Commissioner Amanda Pearson said: “We continue to fully support the IOPC investigation as they work to establish the facts and try to answer the many questions Mr Kaba’s family and others have around his tragic death.”