We went to DLP via Lille on 12th. Fortunately the TGV we intended to catch was sitting on the opposite side of the platform when we got to Lille so there were no problems but we had previously checked where the Eurostar was heading after Lille. Fortunately for us it was going to Paris rather than Brussels so we'd planned to sit tight when we got to Lille and continue on to Paris. Nobody gets on the Paris Eurostar at Lille so we knew there'd be seats available. You could check on the Eurostar website to see where your Eurostar train is going after Lille and hope it's Paris! (When we went Eurostar services were unaffected by the strike).
Coming back we couldn't get from DLP to Lille so we got the RER to Gare du Nord in Paris and went to the Eurostar offices there. They quickly changed our tickets at no extra charge so we went from Paris to London on the Eurostar and got home only 45 minutes later than planned. I was very impressed by Eurostar on this occasion (which has not always been the case!)
So basically, first of all check if your Eurostar is going on to Paris after Lille and, if so, sit tight. Not sure what to recommend if it's heading for Brussels, I'm afraid, though, as Zinderella said, the trains are still running on a much reduced service so you will eventually get there but you may have a long wait. RER services do not seem to be very much affected - or at least they weren't last week.
Coming back, if you have problems getting to Lille, get the RER to Paris and they'll change your Eurostar tickets for you - the return journey should not be problematic.
Good luck!