05AB1E, 14 bytes

ZLãʒO%yн@εNQ}ß

Input as a pair of lists \$[d,e]\$.
Outputs all valid \$[m,n]\$-pairs within the range \$(1,1)\leq(m,n)\leq(\max(d\mathbin\Vert e),\max(d\mathbin\Vert e))\$. If there are possible test cases for which the output would be outside this range, I can fix it. EDIT: Confirmed by @l4m2 that at least one valid pair can always be found with \$m\leq\min(e)\$ and \$n\leq\max(e)\$ in affect.

Try it online or verify all test cases.

Explanation:

Z              # Push the flattened maximum of the (implicit) input-pair of lists
 L             # Pop and push a list in the range [1,max]
  ã            # Pop and take all pairs of this list
   ʒ           # Filter this list of pairs [m,n] by:
    O          #  Sum the current pair: m+n
     %         #  Modulo all inner values of the (implicit) input by this (m+n)
      yн       #  Push the first item of the current pair
        @      #  >= check on each inner value: i%(m+n) >= m
         εNQ}ß #  Check if the first list only contains 0s, and second list 1s:
         ε     #   Map over this pair of checks:
          NQ   #    Check for each check whether it equals the 0-based max-index
            }ß #   After the map: push the flattened minimum to check all are 1
               # (after which the list of valid pairs is output implicitly)

R, 65 63 bytes

\(d,e){while(any(d%%T>=F,e%%T<F)){h=F>T;F=1+F*!h;T=T+h};c(T,F)}

Attempt This Online!

Outputs m+n, m.

JavaScript (V8), 83 bytes

x=>g=(y,i,h=(q,a)=>n=Math.min(...q.map(t=>t*a%i)))=>h(x,-1)+h(y,1)>0?[n,i]:g(y,-~i)

Try it online!

1 layer try

Output m+n, m

Python3, 132 bytes

def f(d,e):
 q=[(1,1)]
 for m,n in q:
  if all(i%(m+n)<m for i in d)and all(i%(m+n)>=m for i in e):return m,n
  q+=[(m+1,n),(m,n+1)]

Try it online!

Charcoal, 21 bytes

→Ｗ¬›⌊﹪ηⅈ⌈﹪θⅈＭ→Ｉ⟦ⅈ⌊﹪ηⅈ

Try it online! Link is to verbose version of code. Outputs m+n and m. Explanation:

→

Start by guessing that m+n might be 1. (This won't end well...)

Ｗ¬›⌊﹪ηⅈ⌈﹪θⅈ

Until all weekend days follow all weekdays...

Ｍ→

... increment m+n.

Ｉ⟦ⅈ⌊﹪ηⅈ

Output m+n and the earliest weekend day as m.