As of today, both in case of a no-deal, a yes-deal, or a delayed Brexit, will EU citizens be required upon entering the UK after 29 March 2019 to fill in a landing card? If yes, how does it look like?
According to the UK Border Force:
You no longer have to fill in a landing card. Your passport (and visa if you have one) will be checked at border control. You’ll usually be asked why you’re coming to the UK.
Therefore the question is pretty much moot, as no one will have to fill out landing cards anymore as of 2019, regardless of what happens during the Brexit process.
I'm afraid that this falls into the category of "we don't know at the present." If free movement persists under the Immigration (EEA) Regulations 2016 or successor legislation, the landing cards will not be required. But the timing of the repeal of that legislation appears uncertain in the event of a no-deal departure. If there is a deal, the timing will depend on the precise details of the deal, which are as yet indeterminate. The prospect of a delay, of course, makes the already uncertain timing even less certain.
Once the EEA regulations are repealed, EU-citizen visitors are likely to be required to fill out the landing card, but the cards are also likely to be scrapped entirely (perhaps only after immigration authorities have had a chance to catch their breath in the wake of departure from the EU). In August 2017, the Home Office announced plans to stop using the cards, expecting to implement those plans by the autumn of 2017, but this never happened.