My Eurostar is scheduled to arrive 30 minutes late in London, and I have an advance ticket for another train to catch, but I'm almost sure I'm going to miss it. What should I do in this case?
In such situations, it is possible (and recommended) to contact the Eurostar personnel during the voyage to have them stamp and sign your ticket to use it as proof. They even announce it ("our personnel will be passing soon, in case you have any questions...").
Then, if you do miss your train due to this delay, you can show the stamped ticket to the rail company staff and have your ticket amended (e.g. if there's a later train to the same destination).
Note: I'm posting this because it just happened to me and I didn't think of asking the Eurostar staff before leaving the train, so when I arrived 5 minutes late for my UK train, I went to ask the rail station staff about it, and only because they were especially forgiving (and after a small sermon) I got it amended. Otherwise, I would have had to recontact Eurostar and spend lots of time discussing with them to have it fixed.
The worst part is that Eurostar's arrivals time website is completely inaccurate: it listed my train as having arrived 30 minutes earlier than it did, so I couldn't even use it as proof that I was not lying about my train being late.
It's a bit too late for you now, but for future reference - Ideally you want to be booking you UK train ticket as a special "CIV" ticket:
CIV is a set of rules shared by European rail operators to protect international journeys by train
You can buy special CIV tickets from your local ticket office (ask for a ticke to London International and show them your Eurostar ticket / Rail-Sail ticket), from Loco2, and a few other options as covered in this Man in Seat 61 article
Assuming you have a CIV ticket, then if your Eurostar is late, then you're entitled to get on a later UK train than booked. It's best to ask the Eurostar staff to stamp your ticket with the magic CIV stamp, to reduce the amount of explaining you have to do, but you're covered. (See this question for what to do if your Eurostar is delayed the other way, and you'll miss your French SNCF train)
Otherwise, if you were careless enough to buy a normal UK train ticket, as the other answer explains, ask the Eurostar staff to annotate / mark your ticket to detail the delay, and hope the other train company is feeling helpful...