I believe that this is all a matter of interpretation. If you have 1 TS aircraft with a BS aircraft in their 12 o'clock, but 5 miles away and reciprocal, I'd give traffic information to the TS aircraft then and there. I'd then update this traffic information a short while later, possibly asking the pilot to report visual. If still not visual by ~2miles, I may pass traffic information to the BS aircraft and ask if they are visual with the TS aircraft. This is not really to assist the BS aircraft, but if the BS aircraft if visual with the TS aircraft this may make a difference later on.
If the TS aircraft still has not seen the other aircraft when range is ~1 mile, I'd be thinking about
suggesting an avoiding action. This is quite a subjective thing, though; if I thought that there was a risk of collision then I'd definitely pass the suggestion avoiding action. However, if the BS aircraft was visual with my TS aircraft then I'd not bother with the avoiding action suggestion.
As far as Deconfliction vs Basic, then I'd avoid the BS aircraft with the DS aircraft and try to coordinate.
BS vs Bs = give traffic information if I thought there was a definite risk of collision only, subject to my workload. Bit more difficult to offer avoiding action here as identification issues are added to the mix.
Some controllers may do this slightly differently, but this is my interpretation of the Duty of Care mentioned by FBW & Vortex. We have a duty to try and prevent collisions, which is why sometimes pilots will get elements of a service above that which they are in receipt, i.e. traffic infromation to a BS aircraft or deconfliction advice to a TS aircraft. As Barnaby has said, it is not the responsibility of the ATCO to offer an upgrade in service, but where safety is likely to be compromised it is part of the Duty of Care to assist where possible.
All this, of course, is subject to workload (that old chestnut, but very pertinent), and does not absolve the pilots of their ultimate responsibility to see and avoid other aircraft OCAS.
HTH
