If the controller is using a primary radar and tracking an air vehicle using skin paints then information may be given in relation to any other air vehicle that is transponding or not. In the case of non-transponding targets that may not provide a good skin paint there may be late warning of traffic.
Information based on skin paints may also be limited when the air vehicle is flying near heavy clouds. Warnings or information may be given in the absence of known height information. If the subject aircraft has no Mode C but a declared flight level then warning may be restricted to Mode C equipped aircraft that present a confliction.
If the conflicting aircraft does not have Mode C then avoidance or information may be given even when there is a large height separation. Pain in the butt springs to mind.
When you talk of SSR failure and mid-games this implies loss of SSR with the subject and not with a stranger. Depending on the type of airspace I would imagine that the game could continue. If the airspace was closed to not participating aircraft and the height of the subject aircraft was known to the controller then there should be no problem. If the airspace was open, and a third party controller needed height information on the subject aircraft, then depending on inter-unit coordination th game might continue or not.
In some airspace the carriage of a serviceable IFF and/or Mode C is essential. In that case it is game, set and match.
As for SSR being sold as IFF, it is a generic term and Mode 4 certainly means IFF capability. Certainly they would not sell IFF to BA.
Health warnings apply as I haven't checked the Mil AIP.