Hi Stephanie,
I suspect in this instance "Flying Squad" isn't an official Police term, but is maybe an unofficial and perhaps grandoise term of the time describing the local Specials or ARP's "mobile" or "quick-response" unit - maybe they were lucky enough to have access to a car and more significantly petrol, which was strictly rationed of course, presumably crewed by a couple of the younger & fitter members of the unit ?. No disrespect at all to Sir Evan Gwynne-Evans or any of those who so ably led ARP, Home Guard and other Civil Defence units during wartime, but it does sound like the term Captain Mainwaring might have used to describe his Bicycle Platoon.

The term seems to have originated in London's Metropolitan Police force
https://en.wikipedia.org/wiki/Flying_Squad

I presume you already have access to whatever archives may still exist for Oaklands Park, or those perhaps held at Gloucester Archives ?

Regarding photos of Special Constables, you may find some in the old newspapers, which in wartime wouldn't carry people's names, such as that shown for Newnham here
http://gloucestershirepolicearchives.org.uk/content/from-strikes-to-vips/wars/second-wo...

You may find references in post-war newspapers such as awards ceremonies or mentions within obituaries and such like, there are a few group photos on the Sungreen site, altho' not for your area.
Search "special constables" here https://www.sungreen.co.uk/Sungreen-Search.htm

For general research of the Specials these may be helpful, seems like any records of the time may not have survived due to wartime pulping, again Glos Archives and the Glos Police Force should be able to advise

http://gloucestershirepolicearchives.org.uk
(cannot post the precise webpage, but search for "special constables")
Sadly the extensive lists of names of police officers within this excellent site only include those serving prior to 1919.
These forum threads may help you, for example to consult the records of magistrates courts when new officers were sign-in.
http://www.open.ac.uk/arts/research/policing/sites/www.open.ac.uk.arts.research.policin...
http://www.oldpolicecellsmuseum.org.uk/page_id__1065.aspx
https://www.british-genealogy.com/threads/71359-Special-police-constable-ww2-records
http://gmic.co.uk/topic/165-special-constables-war-reserve-pcs/

I hope this may be of some help, and wish you good luck. As someone who's tried researching the local Home Guard, I suspect records will be few and far between.
atb Jeff