I'm sorry if this has been covered elsewhere, but all the related questions I found asked the opposite scenario (using the tool you own without having proficiency), and the PHB is pretty poor in explaining how these tools work.
I'm gonna use as an example the Cartographer's tool:
Using cartographer's tools, you can create accurate maps to make travel easier for yourself and those who come after you. These maps can range from large-scale depictions of mountain ranges to diagrams that show the layout of a dungeon level.
Components. Cartographer's tools consist of a quill, ink, parchment, a pair of compasses, calipers, and a ruler.
The way some of the skills benefits are described seems to imply that you would get the bonus from the proficiency to your ability checks, even if you don't have the tools in your equipment, because it would be granted by the experience/knowledge you acquired from using them, and not by something the components themselves allow you to do, for instance:
Survival. Your understanding of geography makes it easier to find paths to civilization, to predict areas where villages or towns might be found, and to avoid becoming lost. You have studied so many maps that common patterns, such as how trade routes evolve and where settlements arise in relation to geographic locations, are familiar to you.
I don't see why someone would suddenly not be able to recognize "common patterns" that they are familiar with, only because they don't have the tools at hand. A similar logic would apply to this one:
Arcana, History, Religion. You can use your knowledge of maps and locations to unearth more detailed information when you use these skills. For instance, you might spot hidden messages in a map, identify when the map was made to determine if geographical features have changed since then, and so forth.
Actually, it seems that owning the tools is only necessary for actions outside of ability checks, proper/unique to their components, for instance:
Craft a Map. While traveling, you can draw a map as you go in addition to engaging in other activity.
Because you can't possibily craft a map without the tool's components, the same logic could be applied to the other activites, suggested at the tool's table:
|Determine a map's age and origin||10|
|Estimate direction and distance to a landmark||15|
|Discern that a map is fake||15|
|Fill in a missing part of a map||20|
So, is there any RAW/RAI answer to this? And how would you as a DM personally rule it?
Thanks in advance!