Notwithstanding the earlier quote from RAF regs (which seems only to exclude foreign badges), ISTR an obscure mixture of regs from earlier cases.
RAF and Army wings may legally be worn on each other's uniform, but only until unless/until you qualify for the "home" set. Examples I'm aware of include RAF pilots who joined Army and/or TA in non-flying roles (still wear RAF wings) and Army pilots transferring who wear theirs until awarded RAF wings.
The RN (as ever) is different. IIRC, under RN rules, no other (non-RN) wings may be worn on their uniform, nor may RN wings be worn on non-RN (OK pedants, or RM) uniforms.