I have been asked about this badge before as I have an interest in Home Front items and happen to live in the London Borough of Bromley.
Whilst I cannot provide an exact identification I don't think this has anything to do with the British Red Cross. The style is similar to items produced for the American Red Cross - like those shown - especially as even on khaki battledress the BRCS wore blue backed cloth shoulder titles.
A quick internet search shows that there are two US Bromleys - Bromley, Alabama and Bromley, Kentucky. I think it's more likely this belongs to one of those two.
Jon
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