UK passport holders and UK airlines are subjected to differing ( effectively third country status) immigration parking and handling right across the EU as we have never signed Schengen so aboslutely no immediate changes are expected on entering the first point of call into any Schengen signatories via air least not till 2022 anyway,.
There is plenty of time to agree Visa FREE travel betwen the EU and UK to be effective from 2022 onwards imho.
And yes it might require movement back towards Free Movements rules again like it or not .
Every time I fly ( and it’s numerous times a year) into Lisbon it’s always remote parking buses and joining the queue with Brasilians Ukrainians and Angolans . Amsterdam it’s a separate corridor for UK And Eire.
The place that will have to be beefed up is Dover/Calais and there WILL be disruption like it or not.
As for CTA with Eire it’s got nothing to do with Brussels at this time , although I personally think the current Taoiseach may well now look to joining Schengen to benefit Eire if the UK goes Hard Brexit with much of their import/exports directed to Containers and ro-ro ferries to France and Spain being built up over the coming years.
They will need to do this to avoid TIR land bridge paper work and seals on moving goods across a third country.
Pretty much like you do to Greece moving goods over land .