There is a difference in the quotes you are using.
const url = "http://mysite.com:8000/service=206&assignment_group=Enterprise%20Network%20Development&start_date=1453161660";
These double quotes are representing a string. You are telling the compiler to parse as is. And so do the single quotes in the quoted code below.
const url = 'http://mysite.com:8000/service=206&assignment_group=Enterprise%20Network%20Development&start_date=${tomorrow}';
In the last one that “doesn’t parse”, you seem to have forgotten a = sign to separate a key from a value.
If you simply change to one of these to examples, It should work. But there is a change that the spaces in your url (%20) will give you some trouble.
const timestamp = (new Date().getTime()) + (24 * 60 * 60 * 1000);
const url = 'http://mysite.com:8000/service=206&assignment_group=Enterprise%20Network%20Development&start_date=' + timestamp;
const timestamp = (new Date().getTime()) + (24 * 60 * 60 * 1000);
const url = `http://mysite.com:8000/service=206&assignment_group=Enterprise%20Network%20Development&start_date=${timestamp}`;
Note that the last example uses backticks instead of single / double quotes.