You can credit your Eurostar ticket to Eurostar Frequent Traveller, Thalys TheCard or SNCF Voyageur. If you book the tickets on the Eurostar website, and only at booking time, you can also credit to Eurostar Points Plus.

You can't credit to any airline programs, nor is their any status benefits to/from any airline programs.

Note - It's Avios UK = ex-airmiles that you can book Eurostar tickets with, not Avios = BAEC, though of course you can transfer between the two. The Avios UK booking with Eurostar is much like the deal they have with Nectar (shopping loyalty scheme), so don't think that just because you can use Avios there's some magic airline link. BA sold their Eurostar shares years and years ago