Find the right path

Slip, 47 bytes

`a(?,[`-+]*((`/<|`\>)[`|+]*(`/>|`\<)[`-+]*)*`:)

Test it here.

Yay for undocumented features...

Explanation

Slip is basically a two dimensional regex syntax and by default Slip programs print the subset of the input that they matched. In this case I'm simply matching a valid path. To prevent printing the entire path, I'm usng the undocumented (?,...) groups which simply indicate that the characters matched inside should be omitted from the output.

As for the regex, unfortunately, there's some duplication because \ and / need to be treated differently depending on whether we're moving horizontally or vertically. On the plus side, since we know that the path starts and ends horizontally, we know that there's an even number of \ or / in every path, so that we can match two of them at a time.

`a             # Match a letter.
(?,            # Match but don't include in output...
  [`-+]*       #   Match a horizontal piece of path, consisting of - or +.
  (            #   Match 0 or more vertical segments...
    (`/<|`\>)  #     Match a / and turn left, or a \ and turn right.
    [`|+]*     #     Match a vertical piece of path, consisting of | or +.
    (`/>|`\<)  #     Match a / and turn right, or a \ and turn left.
    [`-+]*     #     Match a horizontal piece of path, consisting of - or +.
  )*
  `:           #   Match a : to ensure that this is the correct path.
)

Python, 221 bytes

def P(s,c=""):
 l=s.split("\n")
 v=[0,-1]
 p=[(i,l[i].index(":"))for i in range(len(l))if":"in l[i]][0]
 while c in"-:|+/\\":
    p=map(sum,zip(p,v))
    c=l[p[0]][p[1]]
    v=v[::1-(c=="\\")*2]
    if"/"==c:v=[-v[1],-v[0]]
 return c

The first indention is just one space, in the while loop it's a tab.

Javascript (ES6), 117 104 bytes

p=>(r=d=>(c="\\/ABCD".indexOf(p[x+=[-1,-w,w,1][d]])+1)>2?p[x]:r(d^c))(0,w=p.indexOf`
`+1,x=p.indexOf`:`)

Test cases:

let f =
p=>(r=d=>(c="\\/ABCD".indexOf(p[x+=[-1,-w,w,1][d]])+1)>2?p[x]:r(d^c))(0,w=p.indexOf`
`+1,x=p.indexOf`:`)

var p0 = 'A------#\nB------#\nC------#\nD------:',
    p1 = 'A-\\ /---:\nB-+-/ /-#\nC-+---+-#\nD-+---/  \n  \\-----#',
    p2 = '  /-\\    \nA-+\\\\---#\nB-/\\-\\/-#\nC----++-#\nD----+/  \n     \\--:',
    p3 = 'A-\\        \nB-+\\       \nC-++\\/----#\nD-+++//---:\n  \\++-//--#\n   \\+--//-#\n    \\---/  ',
    p4 = '  /-\\     \nA-+-/-\\   \nB-+-+-\\--#\nC-+-/ |/-#\nD-\\---++-#\n  \\---+/  \n      \\--:';

console.log(p0, '=>', f(p0));
console.log(p1, '=>', f(p1));
console.log(p2, '=>', f(p2));
console.log(p3, '=>', f(p3));
console.log(p4, '=>', f(p4));