R, 98 96 bytes

s=function(m,r,j=choose(1:(m+r),r))if(r)`if`(!m,1:r-1,c(s(m-max(j[j<=m]),r-1),max(which(j<=m))))

Try it online!

Commented:

choose_series=      
s=function(m,r,         # recursive function s
 j=choose((m+r):1,r))   # j = all relevant values of choose(c,r)
 if(r)                  # if r==0 don't return anything else
  `if`(!m,              # if m==0 ...
   1:r-1,               # ...just return the remaining r-series minus 1
   c(                   # otherswise return ...
    s(                  # recursive call to self, with
     m-                 #   new m = current m minus ...
      max(j[j<=m])      #   ... highest value of j less than or equal to m
     ,r-1),             #   new r = r-1;
    ((m+r):1)[j<=m][1]  # appended to the highest value of c for which...
   )                    # ...j is less than or equal to m
  )

R, 98 96 bytes

s=function(m,r,j=choose(1:(m+r),r))if(r)`if`(!m,1:r-1,c(s(m-max(j[j<=m]),r-1),max(which(j<=m))))

Try it online!

Commented:

choose_series=      
s=function(m,r,         # recursive function s
 j=choose((m+r):1,r))   # j = all relevant values of choose(c,r)
 if(r)                  # if r==0 don't return anything else
  `if`(!m,              # if m==0 ...
   1:r-1,               # ...just return the remaining r-series minus 1
   c(                   # otherswise return ...
    s(                  # recursive call to self, with
     m-                 #   new m = current m minus ...
      max(j[j<=m])      #   ... highest value of j less than or equal to m
     ,r-1),             #   new r = r-1;
    ((m+r):1)[j<=m][1]  # appended to the highest value of c for which...
   )                    # ...j is less than or equal to m
  )

(but, frustratingly, my approach here still comes-out longer than an 84 byte port of Arnauld's approach...)

R, 98 bytes

s=function(m,r,j=choose((m+r):1,r))if(r)`if`(!m,1:r-1,c(s(m-max(j[j<=m]),r-1),((m+r):1)[j<=m][1]))

Try it online!

Commented:

choose_series=      
s=function(m,r,         # recursive function s
 j=choose((m+r):1,r))   # j = all relevant values of choose(c,r)
 if(r)                  # if r==0 don't return anything else
  `if`(!m,              # if m==0 ...
   1:r-1,               # ...just return the remaining r-series minus 1
   c(                   # otherswise return ...
    s(                  # recursive call to self, with
     m-                 #   new m = current m minus ...
      max(j[j<=m])      #   ... highest value of j less than or equal to m
     ,r-1),             #   new r = r-1;
    ((m+r):1)[j<=m][1]  # appended to the highest value of c for which...
   )                    # ...j is less than or equal to m
  )

R, 98 96 bytes

s=function(m,r,j=choose(1:(m+r),r))if(r)`if`(!m,1:r-1,c(s(m-max(j[j<=m]),r-1),max(which(j<=m))))

Try it online!

Commented:

choose_series=      
s=function(m,r,         # recursive function s
 j=choose((m+r):1,r))   # j = all relevant values of choose(c,r)
 if(r)                  # if r==0 don't return anything else
  `if`(!m,              # if m==0 ...
   1:r-1,               # ...just return the remaining r-series minus 1
   c(                   # otherswise return ...
    s(                  # recursive call to self, with
     m-                 #   new m = current m minus ...
      max(j[j<=m])      #   ... highest value of j less than or equal to m
     ,r-1),             #   new r = r-1;
    ((m+r):1)[j<=m][1]  # appended to the highest value of c for which...
   )                    # ...j is less than or equal to m
  )