I'm not quite sure why this has turned into such a controversy!
All the photos show the same bike, which is a c1934 OEC, fitted with an engine that was branded OEC but was made elsewhere. 33d6 has explained that the engine came from Matchless, with a custom magneto drive case for the customer. When Matchless used the motor it was a "sloper".
As I pointed out above, the 250 and 500 cc variants of this engine are different in the way that the exhaust ports are arranged on the cylinder head. The smaller version has a larger gap between the cylinder and the port, as shown in the 1934 OEC catalogue. One of the original photos shows the silhouette view, indicating the smaller engine. There was a 350 version that was probably the same silhouette.
The latest photo shows the engine in detail, including the funny exhaust lift mechanisim that 33d6 refers to. It also shows the toolbox high on the back in the OEC position.
A very clear photo from Motor Cycling in 1934 shows a similar OEC (albeit with the optional Duplex steering, and labelled 198cc) with a rego number only 2 digits different from the "mystery" bike. Mini me explains that there is nothing interesting about this because dealers block registered batches of bikes. In this case, a batch of OECs no doubt.
The mystery is solved. The bike is a small capacity OEC, dating from 1934.
Cheers
Leon