#JavaScript (ES6), 178

JavaScript (ES6), 178

F=
r=>r.replace(/#/g,(c,i)=>([d=r.search`\n`,-d,++d,-d,++d,-d,1,-1].map((d,j)=>r[i+d]==c&&(--n,s+=j&2?1:Math.SQRT2),n=1),n||(v=w,w=i)),w=s=0)&&s/2/Math.hypot(v%--d-w%d,~(v/d)-~(w/d))
Yellow=` ### #### 
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Nile=` # 
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Missi=` ### 
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console.log('Yellow River',F(Yellow))
console.log('Nile River',F(Nile))
console.log('Mississippi River',F(Missi))

F=
r=>r.replace(/#/g,(c,i)=>([d=r.search`\n`,-d,++d,-d,++d,-d,1,-1].map((d,j)=>r[i+d]==c&&(--n,s+=j&2?1:Math.SQRT2),n=1),n||(v=w,w=i)),w=s=0)&&s/2/Math.hypot(v%--d-w%d,~(v/d)-~(w/d))
Yellow=` ### #### 
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Nile=` # 
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console.log('Yellow River',F(Yellow))
console.log('Nile River',F(Nile))

r=>r.replace(/#/g,(c,i)=>([d=r.search`
`,-d,++d,-d,++d,-d,1,-1].map((d,j)=>r[i+d]==c&&(--n,s+=j&2?1:Math.SQRT2),n=1),n||(v=w,w=i)),w=s=0)&&s/2/Math.hypot(v%--d-w%d,~(v/d)-~(w/d))

r=>(
 r.replace(/#/g, // exec the following for each '#' in the string
 (c,i) => c: current char (=#), i: current position
 ( // check in 8 directions
 // note: d starts as the offset to next row, prev x position
 // and is incremented up to offset to next row, succ x position
 // note 2: there are 2 diagonal offsets, then 2 orthogonal offsets
 // then other 2 diagonal, then 2 more orthogonal
 [d=r.search`\n`,-d, ++d,-d, ++d,-d, 1,-1].map( // for each offset
 (d,j) => // d: current offset, j: array position (0 to 7)
 r[i+d] == c && // if find a '#' at current offset ...
 ( 
 --n, // decrement n to check for 2 neighbors or just 1
 s += j & 2 ? 1 : Math.SQRT2 // add the right distance to s
 ),
 n = 1), // n starts at 1, will be -1 if 2 neighbors found, else 0
 // if n==0 we have found a start or end position, record it in v and w
 n || (v=w, w=i)
 ),
 w=s=0), // init s and w, no need to init v
 // at the end 
 // d is the length of a line + 1
 // s is twice the total length of the river
 // v and w can be used to find the x,y position of start and end
 s/2/Math.hypot(v%--d-w%d,~(v/d)-~(w/d))
)

r=>r.replace(/#/g,(c,i)=>([d=r.search`
`,-d,++d,-d,++d,-d,1,-1].map((d,j)=>r[i+d]==c&&(--n,s+=j&2?1:Math.SQRT2),n=1),n||(v=w,w=i)),w=s=0)&&s/2/Math.hypot(v%--d-w%d,~(v/d)-~(w/d))

r=>(
 r.replace(/#/g, // exec the following for each '#' in the string
 (c,i) => // c: current char (=#), i: current position
 ( // check in 8 directions
 // note: d starts as the offset to next row, prev x position
 // and is incremented up to offset to next row, succ x position
 // note 2: there are 2 diagonal offsets, then 2 orthogonal offsets
 // then other 2 diagonal, then 2 more orthogonal
 [d=r.search`\n`,-d, ++d,-d, ++d,-d, 1,-1].map( // for each offset
 (d,j) => // d: current offset, j: array position (0 to 7)
 r[i+d] == c && // if find a '#' at current offset ...
 ( 
 --n, // decrement n to check for 2 neighbors or just 1
 s += j & 2 ? 1 : Math.SQRT2 // add the right distance to s
 ),
 n = 1), // n starts at 1, will be -1 if 2 neighbors found, else 0
 // if n==0 we have found a start or end position, record it in v and w
 n || (v=w, w=i)
 ),
 w=s=0), // init s and w, no need to init v
 // at the end 
 // d is the length of a line + 1
 // s is twice the total length of the river
 // v and w can be used to find the x,y position of start and end
 s/2/Math.hypot(v%--d-w%d,~(v/d)-~(w/d))
)

Test

F=
r=>r.replace(/#/g,(c,i)=>([d=r.search`\n`,-d,++d,-d,++d,-d,1,-1].map((d,j)=>r[i+d]==c&&(--n,s+=j&2?1:Math.SQRT2),n=1),n||(v=w,w=i)),w=s=0)&&s/2/Math.hypot(v%--d-w%d,~(v/d)-~(w/d))
Yellow=` ### #### 
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## # # ##### 
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Nile=` # 
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console.log('Yellow River',F(Yellow))
console.log('Nile River',F(Nile))