Now, here is an Engineer's hat.
Royal Engineer visor cap of British Army
Badge: Royal Engineers as noted on the bottom banner.
Leather strap & brass RE buttons.
Visor is cloth over a stiff bill.
This is the soft visor cap that was worn while off-duty and when not wearing the steel helmet. These visor caps were worn by officers and NCO's. It was common to see enlisted wear these in WW1 but I'm not sure about WW2. Usually, the enlisted men wore the folding cap they refer to as "side cap".
The only damage to this is that the plastic liner has melted into the fabric of the top, which helps it keep its shape but discolored the top.
I began collecting British regimental cap badges a long time ago. Then I picked up the pace over the last few years. Now my collection of British, Indian, & Canadian cap badges has topped 100 count.
This is a photo of one of 2 cases that I displayed at my city's library.
British Cap Badges & Hats at local library.
Top: Coldstream Guards- Modern(Black/White), Royal Engineers- WW2 (Khaki)
Bottom: Honourable Artillery Company- 1950's(Black/Red) and WW2 Side Cap
Edited: The khaki hat was referred to as "Dress Hat No. 2". Dress Hat No 1 was the colored hat worn for formal dress or parade occasions. The Coldstream Guards hat and the H.A.C. hat are both Dress hat No. 1. Just as each British regiment had its unique regimental cap badge, also each regiment had its unique dress uniform. They varied in style, colors and buttons. One regiment may wear royal blue and yellow and another black and red. That is why collecting British hats is a fun hobby.
Also, the Coldstream Guards hat may look like it is pulled down over his eyes. It was. The front of the hat was flat--you can lay a ruler vertically from the peak to the brass-trimmed bill and the hat would be flat against the ruler. The bill rested on the nose of the wearer. The Coldstream Guards was one of the 5 foot guards of the Royal Palaces.
Steve