I have been researching this question, and while many articles refer to Sherriff living and writing in Selsey, there is very little said about the actual location of the house. The actual reference which talks about the bungalow leads to only supposition. I have cited the article below, so you can read it for yourself. Here is what I can infer:
RC Sherriff's mother did own a house on East Beach Road called Sleepy Hollow; however it referred to a house rather than a bungalow. Sherriff is also seen in a picture sitting on the beach in 1927. It is known that the family rented various houses throughout Selsey. However, there is no indication that he purchased the house on East Beach Road for his mother. (He did purchase a Rolls Royce.) Since Journey's End was performed in 1929, it is unlikely that the house was purchased after.
As for the reference to the bungalow, it seems this is a totally different property from his mother's. According to the article, the bungalow was used to start writing the play Journey's End and to write the sequel, Badger's Green.
We had taken a bungalow close to the sea at Selsey Bill and I cut loose from everything in London to go down there and work in peace.
Based on the resources I could find, there is no indication that the bungalow and the mother's house are the same. While it is possible that Sherriff purchased the house on East Beach Road for his mother, there are no records accessible to ascertain when the purchase occurred. Because of the family's history of going to Selsey on holiday, pictures are not proof of the timeline of events.
Finally, I found an article decrying the tearing down of Sherriff's house, Rosebriars, in Escher. According to the article it was the house where he lived most of his adult life with his mother until his death. Rosebriars was located on Seymour Road, Hampton Wick (a province of Surrey).
More information may be contained in Sherriff's autobiography, No Leading Lady. Based on the evidence, I would say, no, he did not purchase a bungalow on East Beach Road for his mother after the success of Journey's End.