A key principle of the covenant is that the armed forces community should not be at a disadvantage to the civilian community, but nor does it place them in a position of advantage, either.
For veterans, if you take the example of medical care, that is provided by the National Health Service. They should receive the same service as everybody else in the civilian community.
Where their need for the National Health Service is a direct result of their military service, then indeed there may be some form of prioritization and particular attention to making sure they get that care--ahead of the queue, if necessary, perhaps. But there isn't, if you like, any intent behind the covenant to place veterans at a specific advantage at all. It's to make sure that everybody's treated equally, but veterans recognized for their service given.
Veterans, for example, would receive recognition and gratitude, positive measures to prevent disadvantage, and indeed whatever financial packages by way of pensions or disability pensions and so forth that they may be entitled to.