Strictly speaking it won't, because inspite of what most books say, TCAS doesn't interrogate Mode A.
What you probably mean is a Mode A & C transponder with altitude reporting not fitted or deselected.
In this case TCAS will issue a TA if the flight past is assessed to be a threat horizontally. This assessment is based on looking at closing speed, and the rate it is changing, to estimate the miss distance. If the miss distance is small then the alerting condition is met. (The value for the miss distance depends on closing speed but also bnacked up by an absolute minimum value that is related to altitude / radio altitude)
When the horizontal condition is being met, a TA is generated 20-48 seconds prior to the predicted closest point of approach.
pb