Open Return

A lot of airlines sell tickets one way between North America and Europe, especially the non-legacy airlines like Icelandair, WOW, Norwegian, Aer Lingus, and they have connecting points all over Europe.
I'd make sure you passports have at least a year of validity left on them before you leave. Have some kind of plan on your return. The first question the immigration officer will ask when you arrive is "How long are you planning to stay?" Make it clear you will be out of the Schengen Treaty zone well before your 90 days maximum stay is up. UK is not in Schengen, but has a separate maximum stay length. With just a one way ticket, you may have to show you have the wherewithall to buy a ticket home when the time comes.

I believe the immigration officer in London will ask how long you will be staying in the UK. You should have a definite answer to that.

Norwegian Air and Condor also book flights one way at a time.

Should we just try to get a cheap round trip ticket and then only use
one way and buy another cheap round trip to get home when we're ready?

Price both options to make a decision. Just be aware that 90 days is the cap for Schengen countries (done in a maximum of 180 days) including both arrival and departure dates. Give yourself a few days' cushion in case the flight leaving your last Schengen nation cancels.

These are the UK regulations to meet for you to satisfy the officer for entry that you are a genuine visitor:

(a) will leave the UK at the end of their visit; and
(b) will not live in the UK for extended periods through frequent or successive visits, or make the UK their main home; and
(c) is genuinely seeking entry for a purpose that is permitted by the visitor routes; and
(d) will not undertake any prohibited activities set out in V 4.5 – V 4.10; and
(e) must have sufficient funds to cover all reasonable costs in relation to their visit without working or accessing public funds. This includes the cost of the return or onward journey, any costs relating to dependants, and the cost of planned activities such as private medical treatment.

This shows that strictly speaking you don't need a return or onward journey to meet (e), only the wherewithal but it sure helps to meet (a).

I would call a couple major airlines and ask that question. Few years ago when our son studied abroad his return was very uncertain. We purchased a ticket through United with an open return. Trying to remember the details, but I think he had to schedule the return flight about two weeks out. There could have been a couple of other restriction but just don't remember.

You picked a good time of year to go; you should have no problem finding a deal on a one way ticket back using Wow or Norwegian Airlines, especially in November after Thanksgiving 3 weeks before Xmas
You just need Internet access to book it while over there.

These days there are enough airlines (already mentioned) selling bargain-rate one-way tickets that this will not be an issue at all.

However, for the sake of completeness I will note that folks with frequent-flyer miles can easily do what you are proposing. One-way frequent-flyer tickets on many (maybe all) airlines are obtainable at half the "cost" of round-trip tickets. Twice I have headed over to Europe on a frequent-flyer ticket with no return trip booked. No immigration official questioned that in Rome or Madrid. It probably helps that I'm clearly over 60 years old. I had no trouble finding a return flight about 2 months before my return date, but I was checking periodically to be sure of continued availability.

One could do a mix-and-match trip if one didn't have enough frequent-flyer miles for the round-trip: Travel one way on WOW, Icelandair, Norwegian Air Shuttle or Condor; use miles for the other half of the ticket. That would also be a way to get around a situation in which there were no economy-level frequent-flyer tickets available for one of the transatlantic flights. And it's a technique I may use in the future if I'm planning to return from London, which comes with extremely high taxes on frequent-flyer tickets.

We fly to London on frequent-flyer tickets every year, booked one way at a time as acraven suggests. We have never had an immigration officer at Heathrow ask to see a return ticket or question our plans about that; just how long we are staying in the U.K., purpose of our visit, etc.

On the other hand, when we flew from Sydney Australia to New Zealand on Emirates with a one-way ticket, we would not have been allowed the board the plane had I not brought a copy of our return tickets back to Australia ( booked separately, on Air New Zealand).. Some countries are very particular about this, and the airlines are in charge of monitoring it. But it should not be a problem in Europe.

Good point, Tom. I opted to pay for last year's ticket because it was into Nice and out of London, and I caught a great fare that was well under $600, including a couple of seat-selection fees. Instead, I used not much more than half as many miles for a domestic roundtrip I was forced to make over the holidays that would otherwise have cost about $700.

I think the main thing is, when your passport is being stamped and they ask how long are you staying, your answer should be something other than we don't know. That has a possibility of getting you on the next plane home. They'll want to know that your plan is to stay within your permitted visa-free travel permission.

There's something else to consider. September-Otober is not low season, and it's less "shoulder" season and more high season than decades ago. If you are budget travelers, your "last-minute" options for hotels may be limited and you won't get the best deals. If you are high-end folks, then it's not a problem most of the time. Notable exceptions would be Paris when there's a lot of pressure on all hotel levels in September due to some large trade shows then, and maybe Munich for Oktoberfest. Since you are looking to save money on your long-haul flights, I tend to think you aren't high-enders. Another consideration is that travel is expensive if you choose last-minute. If you buy tickets 2-6 months in advance for high-speed trains and budget flights, you will get huge discounts. Prices only go up.

I highly recommend you read Rick's Europe Through the Back Door and think about the big advantages to planning a big trip in advance. It's possible that you'll end up staying a day or two longer in one place or another than you'd have done with hindsight, but nearly everywhere I've been, I've wanted more time . . . but if I'd stayed longer in one city, I would not have had the time (or $$$) for the next one. I've also learned what my personal travel style is. I am more efficient when I know just how long I have in a place. The one time I had "unlimited" time was a 2-week-plus apartment stay in Paris and I frittered away a lot of that - but for me it was worth it since I love just being in Paris. Most anywhere else, I'd have frittered and missed out.