How to remove the first colon ':' from a timestamp?

To cut off the first character, you can also use cut -c:

$ echo ":29.06.2019 23:03:17" | cut -c 2-
29.06.2019 23:03:17

Use

cut -d: -f2-

instead of

cut -d: -f2

to get anything from the second field to the end of line:

TDS="$(grep 'Logfile started' process.log |  awk '{print $3,$4}' | cut -d: -f2-)"
echo "$TDS"

Here is a sed solution:

$ echo ':29.06.2019 23:03:17' | sed 's/^://'
29.06.2019 23:03:17

What the command sed 's/^://' is doing is substitute s the colon character : from the beginning ^ of each line with the empty string //.

Here is a tricky awk solution, where we changing the field separator to ^:, described above, and output the second field (of each line):

$ echo ':29.06.2019 23:03:17' | awk -F'^:' '{print $2}'
29.06.2019 23:03:17

The task could be accomplished also with grep (explanation), probably this could be the fastest solution for large amount of data:

$ echo 'Logfile started :29.06.2019 23:03:17' | grep -Po '^Logfile started :\K.*'
29.06.2019 23:03:17

Or process the file directly by the following command, where the limitation ^ is removed:

grep -Po 'Logfile started :\K.*' process.log

The above could be achieved also by sed and capture groups ()->\1:

sed -nr 's/^.*Logfile started :(.*)$/\1/p' process.log

Where the expression ^.*<something>.*$ will match the whole line, that contains <something>. The command s/old/new/ will substitute this line by the content of the first capture group (the expression in the brackets could be more concrete). The option -r enables the extended regular expressions. The option -n will suppress the normal output of sed and finally the command p will print the matches.