Normal score transformation
As you suspected, correlations will be heavily changed by such a transformation. The reason for this is also exactly as you described, namely that using ranks rather than values for the transformation eliminates all of the information needed to actually preserve these relationships. As an example, suppose we have a single observation with rank 1 across all observations that none the less shows a decrease in absolute value across them (i.e., there's a correlation with a negative slope over time). Such a transformation will result in a correlation of 0.

One might be able to find a different transformation that preserves this relationship, or might be able to simply pool all of the time points together before computing the ranks (this would then at least partially preserve some correlations, though depending on what's needed to properly normalize the data it may cause other problems), but in general there's no guarantee that this is possible.