One of many ways, try:
Sub TextFile_FindReplace()
Dim TextFile As Integer, FilePath As String, FileContent As String, x As String, m, j, p
FilePath = "D:\MyReport" '<<<adjust
x = Dir(FilePath & "\*.txt")
Do
TextFile = FreeFile
Open FilePath & "\" & x For Input As TextFile
FileContent = Input(LOF(TextFile), TextFile)
Close TextFile
m = Split(FileContent, vbLf)
For j = 0 To UBound(m)
If InStr(m(j), "\") > 0 Then
p = Split(m(j), "\")
m(j) = Join(Application.Index(p, Application.Transpose(Application.Evaluate("row(2:" & UBound(p) + 1 & ")"))), "\")
End If
Next j
TextFile = FreeFile
Open FilePath & "\" & x For Output As TextFile
Print #TextFile, Join(m, vbLf)
Close TextFile
x = Dir
Loop Until x = ""
End Sub
Run it only once, repeating it will progressively remove the first part of each line up to and including the first backslash. It overwrites existing files.
There are 2 instances of vbLf which might need a tweak depending on the actual content of your .txt files.
snb here would probably produce a one- or two-liner