If you book a Brussels to Lille ticket, you will not have to show a passport or ID card. The whole point of the new setup (including the special schengen departure lounge upstairs at Brussels) is that routine checking of passports or ID cards for intra-schengen journeys is not allowed. As such, a Brussels to Lille ticket gets you a different checkin area, a dedicated coach (18) with a security guard checking you don't sneak off, and a special exit at Lille (with security guards again checking you don't sneak off)

If you booked a Brussels to London ticket, and tried to leave the train in Lille, you will need to show a passport or ID card. You'd have left the Schengen immigration area in Brussels, and so would need re-admitting one way or another in Lille. Expect it to be a faff, and lots of questions, as that's not how you're supposed to do it!