It does depend what ticket you have, and who you're with.
If you are delayed on another RailTeam service, the ‘Hop on the next available train’ (HOTNAT) feature means that you may not even need to rebook (depends on the station), but if you do it'll be easy. (Keep the ticket from the delayed service though!)
If you bought a domestic ticket that included CIV protections, eg a ticket to London International (CIV) or similar, when you get to the Eurostar station you need to queue up at the ticket office, show them the ticket for the delayed service, and they'll re-issue you with a new ticket for the next available Eurostar. If the delay means you missed the last train of the night, they'll have to sort you out with accommodation too.
If you don't have a domestic ticket with CIV protections, then phone Eurostar as soon as things go wrong, so they can put a note on your account that you may miss your Eurostar due to delays. Next, try to find someone from the train company who has delayed you to write on your ticket / give you a bit of paper confirming the delay. This bit isn't required, but it does seem to help! Finally, when you get to the Eurostar terminal, queue up at the ticket desk, explain about the delay, show them the ticket for the delayed train, tell them you've already spoken to customer services, and ask nicely. (Whether or not you are already covered by CIV at this point is something that hasn't been tested in court, so no-one's quite sure, ditto for onward connections after). Generally, if you'd left enough time had the delay not happened, Eurostar will re-book you onto the next available service.
If you're delayed getting to the Eurostar station by something other than a train, YMMV, so it's probably best to leave more time for that. If you're getting a long distance Eurostar (eg a once-a-week train to the south of France), get there early as even if they can get you there another way, it'll take longer and not be as good! If you can buy a domestic ticket with CIV protections included, that's the best as you know you'll be fine. Otherwise, Eurostar are generally very friendly and helpful, and when I've been delayed on a train getting to St Pancras they've always helped me out :)