Build up the current line by concatenating new bits on to it until you reach a new line. When you reach a new line spit out the previous line, then set the current line string to the time code prefix for the new line:

use strict;
use warnings;

my $currLine;

while (<DATA>) {
    next if ! /^\\ TC: +(\S+) (.+)/;
    
    my ($timecode, $tail) = ($1, $2);
    
    if ($tail !~ /^÷/) {
        $currLine .= $tail;
        next;
    }
    
    print "$currLine\n" if defined $currLine;
    $currLine = "\ TC:  $timecode $tail";
}

print $currLine if defined $currLine;

__DATA__
\ TC:  16:12:48;28 ÷1426÷1426÷142D÷142D÷1470÷1470AND W
\ TC:  16:12:49;08 HI
\ TC:  16:12:49;11 LE
\ TC:  16:12:49;14  OVER
\ TC:  16:12:49;19  T
\ TC:  16:12:49;21 HE L
\ TC:  16:12:49;24 AS
\ TC:  16:12:49;27 T 10
\ TC:  16:12:50;07 ÷1426÷1426÷142D÷142D÷1470÷1470YEARS
\ TC:  16:12:50;17  W
\ TC:  16:12:50;19 E'
\ TC:  16:12:50;21 VE M
\ TC:  16:12:50;25 AD
\ TC:  16:12:50;27 E
\ TC:  16:12:51;09  MAJ
\ TC:  16:12:51;13 OR
\ TC:  16:12:52;22 ÷1426÷1426÷142D÷142D÷1470÷1470INVESTMENTS
[download]

Prints:

 TC:  16:12:48;28 ÷1426÷1426÷142D÷142D÷1470÷1470AND WHILE OVER THE LAS
+T 10
 TC:  16:12:50;07 ÷1426÷1426÷142D÷142D÷1470÷1470YEARS WE'VE MADE MAJOR
 TC:  16:12:52;22 ÷1426÷1426÷142D÷142D÷1470÷1470INVESTMENTS
[download]