Heathrow to Oxford: Train or Bus?

I've taken the bus between Heathrow and Oxford many times and have no complaints. There can be a bit of traffic on the M25 but once the bus leaves the M25 it has never been any problems.

Speaking of expense, have you looked at the cost of storing luggage at Heathrow? £15 per bag if more than 3 hours.

Where do you go after seeing Oxford?

There are only 2 companies- National Express and Oxford Bus Company.
Both are highly regarded.
Generally people only post reviews because they are dis-satisfied, not because they are happy. Often that happens due to unrealistic expectations, or just because people are moaners.
Why don't you find and prebook luggage storage at somewhere like Stasher in Oxford?
Why do a day trip from Heathrow? Are you on a very long layover?

Like LAX, Heathrow can be hell on earth.

Please reconsider that plan. Instead, land, go through immigration, use the HEX or Elizabeth Line to Paddington and from there to your accommodation. Get unpacked, refreshed and head out to explore the neighborhood where you are staying. Find the closest market, Underground station, cafe, pub.

Do Oxford the next day.

Our thinking was that since we only have a few days in London before we have to make our way to Berlin and ultimately Munich to drop off our 19 year old daughter, it seemed a shorter and cheaper trip to take the bus from Heathrow, even with the extra cost of storing baggage. After spending the day in Oxford, it would be back to London to our hotel. Also, even though we are experienced travelers (and New Yorkers) , I admit that I'm a bit intimidated by the size of London and the various transportation hubs.
Do National Express or Oxford Bus run from London proper? We are booked into a hotel in the Southwark neighborhood of London.

Thank you for all the feedback.

Between Oxford and London you're better off with the train. Faster and more comfortable. Trains don't get caught up in traffic.

However, coming off an overnight flight and then trying to do a day trip to Oxford right off the plane......I'd be willing to bet you'll be sound asleep on the ride from Oxford into London.

You really don't want to be dog legging back to Heathrow for your luggage, Better by far to take it to Oxford then stash it there for the day, then come through to London direct by train.
Cost isn't the factor here, it is convenience on a day when you will be tired.
You don't say when this is but see this thread about current issues on the train line from Oxford to Paddington-
https://community.ricksteves.com/travel-forum/england/oxford-to-didcot-railway-line-closed

To answer your question no Nat Express don't run more than the odd service from Oxford to London, nor do Oxford Bus Company (they used to), but there is a coach service called Oxford Tube- however the length of journey on there is too long for such a day.
Use the train.
From Paddington to Southwark either use Elizabeth line one stop to Bond Street, then Jubilee Line 4 stops to Southwark; or Bakerloo Line (which also calls at Marylebone, the alternative rail route from Oxford) 9 stops to Waterloo, then Jubilee line 1 stop.

No need to go into London, The Airline bus (coach) takes you between LHR and central Oxford on an hourly basis. The trip takes 1.5-2 hours each way, so you could leave in the morning (allowing time for immigration, baggage retrieval and storage, etc.) and come back in late afternoon or even evening. We used this service from LHR to Oxford a few years ago and it was excellent. But we took our bags with us, stayed a few days in Oxford, then took the train back to London, a lot easier than the round trip you're considering after an overnight flight. I'd hate to wear myself out with a round trip to Oxford, then find myself back at Heathrow facing another trip into London before my day was done.

Unfortunately, it doesn't look like you can store bags at either the rail or bus station (Gloucester Green) in Oxford. Maybe one of our UK friends knows of another way to store bags for a day in Oxford. If so, your return trip could be by train right into Paddington, and then taxi to your hotel.

Or, do the sensible thing as others have suggested -- either spend the first night with your bags in Oxford, or do Oxford as a day trip during your time in London. Or save Oxford for another trip when you can give it some time.

You can store your bags in Oxford t one of the Radical Storage locations https://radicalstorage.com/luggage-storage/oxford?ac=532&subac=c:gad-to:en-ct:cy-lt:ml&gclid=Cj0KCQjwmN2iBhCrARIsAG_G2i6q605HLF62GpsYvaLR1z7PzTnhthIlEMBnXJ9jYkjJWr5TmUsMknsaAvWYEALw_wcB&gclsrc=aw.ds. We just did this last week. Note the one nearest the bus station is up one flight of stairs and the train station location does not open until noon. Pre book online.

There is a Stasher location quite close to Gloucester Green - The Organic Deli Cafe and Wholefoods Store.

Carting your luggage up there for just a day trip at the beginning of your stay though is not what I would recommend, but maybe it suits you.

"Our thinking was that since we only have a few days in London before we have to make our way to Berlin and ultimately Munich to drop off our 19 year old daughter, it seemed a shorter and cheaper trip to take the bus from Heathrow, even with the extra cost of storing baggage."

Forget the storing of bags and retrieving them, and your trip becomes much easier.

Start with what Dick said:
"No need to go into London, The Airline bus (coach) takes you between LHR and central Oxford on an hourly basis. The trip takes 1.5-2 hours."

In my opinion, the best idea is to go straight to Oxford, spend one or two nights in a hotel there, then go into London. You really will not have enough time in Oxford, doing it the way you were planning in your original question; not even worth going.
On your arrival day, jet lag will hit you a couple of hours after you get off the plane. You won't feel like running all around Oxford and then hopping on a train to London.

No matter how few days you have in London, I would stay one or two nights in Oxford first. Two nights would be my preference. Give yourself a full day to enjoy Oxford, take a walking tour of the colleges, enjoy the pubs and good restaurants in Oxford.

You will be back to London someday. If you cancel the first two nights you've planned for there, will that still leave you 2 or 3 nights there? That may be enough for now.