It has long been usual practice in the British Army to have medical insignia on removable brassards that allow it to be clearly displayed when needed, but removed quickly for practical purposes- the red cross on white background is particularly difficult to wash at high temperatures as the red will run and turn it pink, so if it can be removed from clothing before washing then that is a big help! In the UK this insignia is usually attached to DPM or olive green brassards, however the British Army also fights in the desert and this example is in desert DPM with a red cross badge and Union flag stitched to the outside:

The brassard has a loop and Velcro at the top to attach to an epaulette, and a piece of Velcro on the rear face to allow it to be wrapped around the upper arm:

The loop makes me think that this may date from around the time of the First Gulf War as the DDPM shirts worn in the post 2001 conflicts had a chest mounted rank tab, rather than shoulder mounted examples so the brassard would not be suitable for wear with the later pattern of shirt. The feel of the fabric supports this theory as it is a heavier weight cotton and ‘feels’ very similar to the uniforms issued for the First Gulf War. This brassard is completely plain, however other examples exist with a set of pen pockets incorporated into the design. There are no labels on the brassard at all, which is not unusual, and the general feel of the item suggests it was made as part of a small privately acquired batch at regimental level, rather than being sourced through the usual military supply channels.