- In Bricklink Studio go to File > Export As > Export as LDraw...
- Open the file in any text editor such as Notepad
- Find the Bricklink part ID you want to replace it. It will be a number followed by a .dat extension.
- If you have multiple identical bricks with different colours, check the colour id in the second column. Bricklink has a list of colour ids for reference.
- Replace the part ID that precedes the file extension with a valid Bricklink part ID.
- Save the file (as a .ldr file)
- In Bricklink Studio open the file with File > Import > Import Model...

In this example I replaced the Green (ID 2) #3003 2x2 brick with a #3941 2x2 round brick.

The rest of the data includes rotation, transformation and meta data which you should leave unless you know for sure how it works. There are no guarantees connections will be valid, as each part has their own unique geometry and fixed pivot point.
In this case I could safely replace these two parts as their geometry is very similar.
0 Untitled Model
0 Name: Block Tower
0 Author: Ambo100
0 CustomBrick
1 4 20.000000 -24.000000 -60.000000 1.000000 0.000000 0.000000 0.000000 1.000000 0.000000 0.000000 0.000000 1.000000 3003.dat
1 14 20.000000 -48.000000 -60.000000 1.000000 0.000000 0.000000 0.000000 1.000000 0.000000 0.000000 0.000000 1.000000 3003.dat
1 2 20.000000 -72.000000 -60.000000 1.000000 0.000000 0.000000 0.000000 1.000000 0.000000 0.000000 0.000000 1.000000 3941.dat
1 1 20.000000 -96.000000 -60.000000 1.000000 0.000000 0.000000 0.000000 1.000000 0.000000 0.000000 0.000000 1.000000 3003.dat
Notes
This may interfere with generated instructions, it is recommended to replace all parts before creating instructions.
Many text editors have a 'Find and replace' feature you can use to replace all parts as each line only refers to one part. I would recommend using regex expressions if you are familiar with them, I have created an example that can be played with here.
Add the following replacement expression in the 'Replace' window on the bottom of the screen. NEWPARTID must be changed to the new part you want to replace with. (3941 for example). Make sure you don't remove the 1 at the start.
$1NEWPARTID.dat
For finding only certain colours you can use this search expression.
((?:\d+\s+)(?:COLOURID)(?:\s.*))(?:(PARTID).dat)
Replace COLOURID and PART ID (14 and 3003 for example).