A list of Boxes and Police Posts in Exeter, together with a history of the Exeter Box and Post network can be found here:
http://www.exetermemories.co.uk/em/_organisations/police-boxes.php
The page includes a photo of the Police Box at Exe Bridge (as posted earlier by Domvar) and the police post located at Paris Street.

The list isn't as detailed as the Met archive list, but is a good starting point should anyone want to start researching the Exeter Box and Post network.

MAIN STREET BEATS   
SIDWELL STREET   St Annes Chapel (two lights, one for Old Tiverton Rd., one or Blackboy Rd)
EASTGATE   Junction of Paris Street and High Street
HIGH STREET   Opposite Guildhall
FORE STREET, EXETER   Junction with King Street
EXE BRIDGE (demolished)   Bonhay Road - Exe Bridge Sub-Station
BUDDLE LANE   Junction Dunsford Road and Buddle Lane
OTHER BEAT AREAS   
THURLOW ROAD   Originally in Commins Road
POLSLOE BRIDGE   Junction Hamlin Lane
PINHOE ROAD   Whipton Sub-Station (since demolished, now Peel Close)
PRINCE CHARLES ROAD   Junction of Mount Pleasant Road
UNION ROAD   Junction Pennsylvania Road
COWLEY BRIDGE ROAD   Duryard entrance
CLOCK TOWER   Queen Street, junction New North Road
ST DAVID'S HILL   Outside Post Office (Red Cow)
PARIS STREET   Junction Western Way and Heavitree Road
FORE STREET, HEAVITREE   Sub-Station x Congregational Church   (now private property)
HEAVITREE BRIDGE   Honiton Road, junction Birchy Barton Hill
MIDDLEMOOR   Sidmouth Road, junction Old Rydon Lane
HOLLOWAY STREET   Top of Quay Hill
TOPSHAM ROAD   Junction of Feltrim Avenue
TOPSHAM ROAD   Sub-Station, corner of Topsham Road and Burnthouse Lane
BURNTHOUSE LANE   Opposite Bradley Rowe School
RIFFORD ROAD   Junction Ludwell Lane by old Gospel Hall
WILLEY'S AVENUE   Junction with Alphington St.
REDHILLS   Isleworth Road by The Barley Mow pub
EXWICK   Exwick Road, junction with Valley Road
ALPHINGTON ROAD   Alongside wall of Cross House
MARSH BARTON   Marsh Green Road

Sharper copy of the same image above with all of the lamp in shot

Looks like a later model much like the later Newcastle boxes. Think this is a police training centre. Exeter 1930s