Buck converter drops only keeps Vout if no traces are connected
Have you read the guidelines/looked at the recommended layout from Analog?
Page 14-15:
https://www.analog.com/media/en/technical-documentation/data-sheets/ADP2108.pdf

This is where you start for good layout practice, learn from the chip maker recommendation.

You schematics are very hard to read, take good notes from this post:
Rules and guidelines for drawing good schematics
Some examples: missing pin numbers on the ADP2108 symbol, incorrect labeling of Enable pin, output capacitor missing reference designator.

However it seems like this circuit should work when connected to your PIC.
ADP2108 footprint pinout seems correct.

Do you mind sharing the part number / datasheet you've used? It could be the pinout was not laid properly, just a common mistake for creating footprints from scratch.
Maybe not a short to ground but maybe back-driving of some sort.



EDIT - Comments on the posted scope captures

The scope captures you posted look very strange, it looks like the regulator is operating in "burst" mode and the "idle" period is pretty long. Burst mode is for very light-load operation where the load is drawing tiny current, this regulator light-load limit is 80mA (under that limit the regulator switches to a Power Save mode according to Analog's datasheet). It is not so surprising it would operate in this mode with just the PIC connected (unflashed MCU current tend to be very low). What looks strange is the fact that the 1.0V seems to dominate the switching regulator circuit.
I think the mystery revolves around solving these 2 problems:


Problem 1. When regulator output load is disconnected, output is measured at 3.3V. When adding a tiny resistive load (470R), the output goes to 1.0V.


Could you please post some scope captures of the same 2 nodes when output is disconnected ? Is the PWM in burst mode too?
Can you try adding a 10k resistor to the output and get the same scope captures? Is the output at 1.0V already?
If not, can you determine what resistive/current load draws it to 1.0V?


Problem 2. Why can't the regulator regulate over 1.0V as soon as it is loaded?


If problem 1 did not exists, I would have immediately call the Analog part number, because they do have a 1.0V option (ADP2108AUJZ-1.0-R7).
However, seeing 3.3V at no load confirmed the part you ordered is the correct one.

In CH2 trace (blue, output), you can see a tiny voltage bump when the high-side FET is on (eg. L1 is connected to VIN) but the regulator quickly shuts it down and the bump fades back almost immediately, output goes back to a "hard" 1.0V instead (eg. I say "hard" because it looks pretty clean 1.0V step when regulator isn't switching). Is your board exclusively powered by the battery voltage when testing the regulator? If there is another potential power source (even USB) please disconnect.

Does the output stays at 1.0V when you disable the regulator (eg. when Enable pin is low)?