I'll be honest there is so much assumption, half truth and conflation of a number of different issues within this (understandable as you're not that close it and wouldn't know accurately) it's hard to answer without giving you an essay.
Capacity doesn't exist within local structures, so you can manage outbreak and low level testing 'in house' comfortably, but not necessarily mass testing because you need testers, lab capacity, admin function to deliver it. Deloitte have have a hand in various bits of the logistics part of the Government response as well as elements of testing - including coordination of testing centres that are producing Pillar 2 data. As I mentioned regarding the data sharing they don't have the same obligations as a private company as public Sector organisations might because in emergencies they aren't privy to the same legislation as those that fall under the Civil Contingencies Act.
You can read more about that here:
To be used in conjunction with the non-statutory Emergency Response and Recovery and statutory Emergency Preparedness guidance.
www.gov.uk
If Deloitte were passing the data to the NHS, there would be no reason that wouldn't get to PHE and locally to DPH, but it hasn't. Locally that information hasn't been available, as I've been saying in this thread for a few months.
Lots of data in subject to confidentiality so cannot be shared where it is available, but it has only really been available for a few days (normally where you find a aggressive denial of something from MHCLG one day, the information will likely have been received the day before!).
But fundamentally, as per the email response I shared yesterday from Nadine Dories there is no requirement for Deloitte to share their information with PHE or Local Authorities.
Notwithstanding that, where data is available it's not very good, as I've also been saying for months. You can have a scenario, as has been the case across the North West, where at the same site where you have P1 & P2 testing, you can get the P1 data but not the P2.
Edit: if you need to quickly scale something up to hit a national target of let's say 100,000 tests, which you can hold up as a memorable moment in your response to Covid-19 and receive positive headlines and praise for, does it matter if you actually dont do much with that data?