A nonstandard singularly perturbed linear time-dependent control system with multiple point and distributed small delays in state variables is considered. Two much simpler parameter-free systems (the reduced-order and the boundary-layer ones), associated with the original system, are introduced. Due to the nonstandard nature of the original system, the reduced-order one is descriptor (differential-algebraic). A connection between the properties of controllability of these systems and such a property of the original system itself, robust in the small parameter of singular perturbations, is established. A criterion of the impulse-free controllability with respect to the slow variable of the reduced-order system is obtained, and the invariance of this criterion to an auxiliary gain matrix is shown.