It’s doable.
But you will not thoroughly see each country, which IMHO is okay.
This is very basic, but you could take this idea, then look at an atlas:
Start in Ireland, loop around to coast or city with airport, fly/ferry to England, then make a broad loop south, seeing what you want to see in Germany, Italy, and France. You could fly open jaw, or just ferry/fly back to Ireland.

instead of countries, think cites/towns
In 21 days i’d pick maximum! 7 places to go visit
with every transportation segment, you lose exploring time being in transit

Hi, my wife and I backpacked last September – Oct (Sept 10 – Oct 8) and we visited:

London (arrived Sept 10 – left Sept 15)
Edinburgh (Arrived Sept 15 – left Sept 18)
Dublin (Arrived Sept 18 – left Sept Sept 22)
Paris (Arrived Sept 22 – left Sept 27)
Rome (Arrived Sept 27 – left Oct 1)
Naples (Arrived Oct1 – left Oct 3) – mainly a jump point for 1 full day in Capri
Bari ( Arrived Oct 3 – left Oct 7)

The advise posted earlier is dead on, look at it as cities, air travel in Europe is cheap and we flew everywhere, except in Italy (we used the train – and ferry to Capri of course)

Your destinations may be rushed a little, and remember to leave your self enough time to enjoy each place, for example we rushed through Edinburgh and now I wish we had taken an extra day or so there. The city is great and we spent 1 entire day in the Scottish Highlands and up to Loch Ness – if you have time I recommend it.

London, Paris, Rome all these big cities required at least 3-4 days if not more if you want to see the main touristy spots and explore the surrounding areas.

London – we spend 1 day at Windsor Castle and Stonehenge
Dublin – 1 day trip out to the cliffs of Mohr
Paris – 1 day to Mount St Michael
Rome – 1 day to Pompeii

Good luck with your travel plane, I would advise, checking your list and seeing where you want to go the most and why, and use that as the basis of planning your trip, have your must see destinations accounted for and plan around them or in-route to them.
Later
B

Most of the time, the bad weather starts in September. Indian summers are a rarity in the north, so plan for rain and temps that might look like an average of 15 Deg C in the UK and Ireland. Paris can have Indian summer. The weather will be nice in Italy.

Air travel in Europe cheap? Only if you book the no-frills carriers way in advance. Things have changed, but check this out on cheap flights.

Late September and early October are end-of-season in Greece, but the weather is nice. Party places are dead, and the attendees are mainly older. Paris and Rome are still crowded, but nothing like July-August.

In Italy and Southern France, expect fantastic weather for September and the first half of October.

Try to hit one of the weekend wine festivals on the Rhine River, which isn’t far from France of course – St. Goar on 9/20 (amazing fireworks displays that day, fest all weekend) and Boppard on the following two weekends (also w/fireworks on Sat.)