The short answer is that this entirely depends on how you have configured the network dataset. I will attempt a longer answer so you can hopefully get this figured out.
A network dataset is configured with cost (also called impedance) attributes that determine the cost (like distance or time) to traverse each road or intersection. From your question, I presume your network has at least one cost attribute that calculates distance in units of kilometers and another that calculates drive time with units of minutes. Whoever created the data presumably set those up, or else you used ArcMap's network dataset creation wizard, and it configured them for you automatically.
Each cost attribute in a network dataset must be configured so it knows how to calculate the cost for each edge or junction in the network, and it does this using something called evaluators. There are many different types of evaluators, and each of them calculates cost in a different way. The simplest ones just use a constant (every road just uses a fixed value) or a field (just read the cost from a field in the source data). There are other more complicated ones that allow you to use python or VBScript or draw information from traffic or public transit tables. So, what you need to do is to look at the evaluators for your cost attribute to see how they are currently calculated. This will tell you if/how you need to update the fields for your new data. My guess is that the network is using a field evaluator and drawing information from the shape length of the streets, but that might not be the case.
Here is some documentation to help you understand network attributes generally and cost attributes specifically:
ArcGIS Pro:
Network attributes—ArcGIS Pro | Documentation
Cost attributes—ArcGIS Pro | Documentation
ArcMap:
Understanding network attributes—Help | Documentation
