What is the likely impact on flights of grounding an entire aircraft series?

In the short term, obviously, cancelled flights and re-routing on other flights.

In the slightly longer term, it is worth knowing that there are huge numbers of retired aircraft in “graveyards”. Parked up, in desert-y places, awaiting an event precisely such as this. These range from ancient near-junk kept as spare parts, up to fully serviceable aircraft that have recently been retired because the economics of operating them became inferior to the economics of operating new aircraft.

Such as the 737 MAX.

So I would expect that airlines will take out some leases on planes that were recently retired, to fill the gaps in their capacity left by the grounding. Chances are it’s indirect, that some fairly up-to-date airliner is leased by a Modern western airline,
and some other airline elsewhere in the world happily pockets a windfall profit and spends some of it on leasing the retired aircraft. Or maybe, just stiffs its passengers, in parts of the world where consumer rights are next to non-existent.

I don’t know how this works with pilots. How much work is needed for pilots to re-qualify to fly the older jets? I would guess that if pilots have flown the old model earlier in their career, it’s not very much of an issue.

At which point things are back to normal for passengers, apart from flying in some tattier old airliners in place of shiny new ones.

Lots of lawyers will be busy. Boeing may be on the hook for a lot of damages.

In passing, I have always felt happier flying in a well-used airliner than in a brand-new one. The recent sad events, are why.

Depends heavily on the airline. Some airlines operate a single type exclusively and would be effectively shut down, prevented from operating entirely.
Think RyanAir if the 737NG series were to be grounded.

Other airlines may be heavily inconvenienced but still able to operate at reduced capacity.
Emirates would be in that situation if the A380 or 777 series were to be grounded.

Other airlines again would only lose a limited amount of capacity and probably be able to rather rapidly compensate for that through short term wet leasing of additional assets.
That’s where you’d find for example KLM if the 747 series were to be grounded. They’d lose some income mainly because of the increased cost of operation incurred by wet leasing aircraft while still having to pay your existing crew and parking fees as well.

And that’s of course (as already pointed out) just the operational problems. There’s marketing problems as well.
Think KLM’s corporate image which for decades depended heavily on the reliability and majesty of the 747. Were a major flaw in the 747 to lead to grounding of the entire fleet, KLM’s reputation would take a nosedive they might not easily recover from.

Airlines do not make money easily, and losses are very common due to many reasons, crashes or bans are one of them. Hence their plans usually include some scenarios like this. I am sure that their ERP teams (emergency response teams) are dispatched as we speak to solve many problems, including media, recovery plans, etc.

The most logical and feasible solution in case their fleet cannot cover the demand for a short period ban would be some sort of an agreement with another airline to handle the passengers until things get sorted out. This can be achieved quickly and does not require a long term contract and will not result in loosing customers as they would be able to fulfill the great promise of moving them from point A to B safely and on time.

Wet or dry lease (as mentioned in another answer) require a lot of preparations, including certification and training, it’s not like hiring a car, it’s a very long process and usually the contracts are long enough which makes it not the best option for this case which I assume it won’t take long.

FYI, many of IATA training courses which are taken by almost all airlines include scenarios like this. Each airline has an ERP team from different departments that gets activated in such cases to do their pre-defined roles. Their main goal is to get out of such situations with the least possible damage in terms of reputation (the most important impact for long term) and to find options and solutions that would minimize the short term financial impact, such as the aforementioned temporary agreements.

One of the options for affected airlines is to lease aircraft as an interim measure.

There are quite a few companies that have aircraft available to fill in as replacements. They operate either on a “wet lease” basis (including crew) or on a “dry lease” basis (without crew). You can even find an A380 if you want.

Availability varies a lot based on the time of the year, so they could have planes available now, while in the summer it may be a lot more difficult (as many of those places are used for charter flights during that period).

Higher-end airlines on long-haul routes usually do not like to use those much, as they are often second-hand planes, and their interior equipment may be quite different from what they usually provide to their customers (often more geared towards the charter business than business travel), but if that helps avoid cancelling dozens of flights, it’s better than nothing.

There are 2 impacts, one if financial.

It costs money to ground planes (and do full inspection and certification again?) and replace them with other existing planes; not just physically have different planes, but juggling the different range that those planes have and put them on appropriate routes.

For example, Air Canada re-route passengers from to Montreal or Toronto to board different planes that can fly out to the UK (for example)

“In a statement, Air Canada says it is rebooking affected passengers through Montreal, Toronto and Ottawa. Affected customers are asked to contact Air Canada Reservations to change their flights free of charge.”

https://globalnews.ca/news/5047559/boeing-737-max-8-ban-halifax/

The other impact is legal.

If it is found that there is a flaw in the plane system (hardware, software) or in the training related to new system, then a large civil suit can (and will probably) happen.

There are no airlines that fly only this airplane. So for example, Air Canada has cancelled all its flights from Halifax to London, which use that plane. It’s re-routing passengers through Ottawa, Montreal, or Toronto, where the flights use different planes. Some people will cancel, since they didn’t allow for this kind of re-routing delay, and the rest will be absorbed into significantly fuller planes for a few days.

On the financial front, both Air Canada and WestJet have suspended their financial forecasts, suggesting the impact will be significant. Not on flights, but on the companies as a whole. In that article we learn that WestJet

expects to preserve about 86 per cent of passengers bookings on Max flights and find alternative planes for about 75 per cent of Max flights.

I guess the others will simply not happen. Note also that Max was supposed to be very fuel efficient, so switching to other planes might mean the airlines spend more on fuel than they had planned.