England-Ireland-Scotland

It's up to you, but I'd skip Ireland and Scotland and find another place closer to visit in England. Forget the rental car and take a train. Have you booked flight tickets? You could do fly into London and out of Manchester and do the whole trip by trains.

One suggestion, fly into London for a few days, train to Liverpool, day trip to Chester for some Roman history, and a couple of days in Manchester. You'll see a nice sample of England without any awful travel. You'll also like the lower prices in Liverpool and Manchester as well.

Or choose Scotland OR Ireland.
2 or 3 days in each of 2 countries seems pointless.

I spent 10 days in Scotland and barely got a taste. Scotland and Ireland are bigger and slower than you might expect.

We love Scotland, especially the highlands, and have visited there often. However, late October is probably not a good time to go there because you will not have much daylight.

I agree with the above replies that if you have 7 days, you should pick one: England, Ireland, or Scotland. Particularly, as pointed out above, because your daylight hours will be relatively short. Remember that you lose much of day getting from Ireland to England or Scotland. Sure, the flights are short, but you have to get to and from the airports. If you try to see two of these, you will spend too much time in transit and not enough time sightseeing.

With only 7 days, you will want to fly into London and out of your "other country." Since you asked where to fly home from, it sounds like you have flexibility (good that you aren't locked into flying home from London). Depending on your home airport, you may have a nonstop flight home from Manchester, Glasgow, Edinburgh, Dublin, Shannon or Belfast; or, you may need to change one or more times. If you're changing anyway, you may be able to use other airports just as easily (say, Inverness for northern Scotland). If you let us know your home airport, people can offer flight possibilities. To learn who flies where from a particular airport, check that airport's Wikipedia page.

I think Vickie was saying they'll be in London at least 5 days, and then plan an additional 7 days for Ireland and Scotland.
I would encourage you to do exactly what you really love to do, have fun, but you may find that note about there being little daylight at that time of year, it might be something that you want to consider if you have many outdoor sights you want to see there. Definitely do some mapping on like Google Maps or something to get a sense of the time and distance in between destinations. London is quite far south of Scotland and Ireland, so you may want a flight to Dublin or something. Mostly I hear about people taking the train to Scotland.
York is on the way, so that might be a place to take a train to, stop for a night, and continue on. I guess it depends on what you would like to see the most. :)
I know some friends who mentioned that they took a short flight to Ireland & that went OK, but I think in October that's also the month that Brexit is supposed to become more official, so you may want to watch the news from the UK more often. It could effect Irish travel. It's complicated still.

I would suggest reviewing the previous posts on the boards for Scotland and Ireland travel posts, I know I've seen them go into loads of detail about the train lines and all; but there's so much in London, I don't really go anywhere else when I can help it :)

I've also seen the Rick Steves pages & postings on Facebook to be helpful to connect with other travelers.

Probably hiring a guide in the really historic sights would be good, there can be a lot of local, historic details that would ellude a traveler.
Have you watched Rick Steves' episodes for these areas? Some of the guides in his shows (who are also in his books and available for hire) are a real cast of characters :) You Tube also can have a lot of tour and travel videos that are short and informative, by guides in those areas, who will end the video by telling their web site address and you can hire them if you like them.

I find that if I am a member of any clubs or organizations here, that also have locations there, that's a valuable way to build contacts that have helped me when I get there. It's good that you've got some time and can make some connections before you go.
There's a garden museum across the river from Westminster, if you like nature and hidden gems. :) I've lived in London for 2 months in the past, and really didn't see that much of what it has to offer, it's huge and has everything. I may return in sept for grad school, I'll keep making lists of things to do there. :) I'll never see it all, but I can try. Sky Garden's on my list for next time, looks like a good option for a rainy day.
Have fun!
-Alison

On 7 days after London, you should skip Scotland and Ireland or you will waste a lot of time traveling. Save them for another trip.

We did a 28 day drive tour of Wales and England and didn't spend any time in London, except picking up our rental car.

Here is my review of our great trip, with details on hotels, tours, etc.
28 days in Britain and Celebrity Eclipse home
https://www.cruisecritic.com/memberreviews/memberreview.cfm?EntryID=599139

Like the others, with only 7 days, pick one area of one country to visit. Since you’ll already be in London, consider southeast England, or head west towards Bath and the Cotswolds.

I understand it is your first trip and you want to see everything. Please understand with the time you have it will only be the highlights.

Here is what I would suggest if you inisist on seeing both Scotland and Ireland.

After London:

Day 1) Take an early morning train to York. Drop off your bags and spend the rest of the day seeing the town.

Day 2) Take an early morning train to Edinburgh. Drop off your bags and the spend the rest of the day seeing the city. (The hop on hop off bus gives you a guided tour of the city and allows you to get on and off as you please.)

Day 3) Continue seeing Edinburgh and perhaps Stirling Castle.

Day 4) Take an escorted one day trip into the Highlands (Rabbies is a highly rated tour company for this. They use small vans rather than large coaches.)

Day 5) Take a morning fight to Dublin. Spend the day seeing the sights.

Day 6) More of Dublin or a trip outside the city (Perhaps a day trip to Belfast. There are direct trains from Dublin.)

Day 7) Fly home from Dublin if possible or connect back to London.

This is a limited tour of what is available. Believe you will return and fill in the rest. For this kind of trip a rental car is not necessary. Taking escorted one day tours will allow you to see more with a guide.

You might also want to look into a day trip or two out of London.

I like most of Frank's plan. However, not being a huge fan of smaller cities, I would suggest replacing Day 3 with more of the Scottish countryside - perhaps you could rent a car and explore the Borders region AND Day 6 with exploring more of the Irish countryside. You could either head south of Dublin and check out the Glendalough area or head north and check out the Newgrange area. Even better for Ireland, fly instead to either Shannon or Cork and explore the southwest corner of Ireland. Fly back to London from that area.