Vyxal 3 -Ṡ, 14 bytes

Ꮬ[]⟒⑵↻Ꮬ],⨪r1ri

Vyxal It Online! Ṡ flag doesn't eval inputs which is nice for string representations of lists

Explanation:

Ꮬ[]⟒⑵↻Ꮬ],⨪r1r⎇fj­⁡​‎‎⁡⁠⁡‏⁠‎⁡⁠⁢‏⁠‎⁡⁠⁣‏‏​⁡⁠⁡‌⁢​‎‎⁡⁠⁤‏‏​⁡⁠⁡‌⁣​‎‎⁡⁠⁢⁡‏⁠‎⁡⁠⁢⁢‏⁠⁠⁠⁠⁠‎⁡⁠⁣⁣‏⁠⁠‏​⁡⁠⁡‌⁤​‎‎⁡⁠⁢⁣‏⁠‎⁡⁠⁢⁤‏⁠‎⁡⁠⁣⁡‏⁠‎⁡⁠⁣⁢‏‏​⁡⁠⁡‌⁢⁡​‎⁠‎⁡⁠⁣⁤‏⁠‎⁡⁠⁤⁡‏‏​⁡⁠⁡‌⁢⁢​‎‎⁡⁠⁤⁢‏⁠‎⁡⁠⁤⁣‏‏​⁡⁠⁡‌⁢⁣​‎‎⁡⁠⁤⁤‏‏​⁡⁠⁡‌­
Ꮬ[]               # ‎⁡String `[]`
   ⟒              # ‎⁢Left fork doesn't pop the top of the stack here
    ⑵↻    r       # ‎⁣replace `][` with 
      Ꮬ],⨪        # ‎⁤bracket palindromised `],` -> `],[`
           1r     # ‎⁢⁡replace all `[]` with 1
             ⎇f   # ‎⁢⁢flatten `[]` into a list
               j  # ‎⁢⁣join those by the inner part
💎

Created with the help of Luminespire.

tldr - left fork regex -2 because I was looking through documentation for new stuff and noticed that i also just encloses stuff so this works out nicely

Python, 236 155 bytes

def r(o):
 v="[";s=a=n=0
 for c in o:
  if c=="(":
   if s<1:a=n+1
   s+=1
  else:
   s-=1
   if a==n:v+="1,"
   elif s<1:v+=r(o[a:n])
  n+=1
 return v+"]"

Is this a good solution? No. But I don't think there'll be another answer here soon, and I'm very new to codegolf so I thought I might try my luck here :p

Attmpt this online!