Q:

Heathrow to Oxford without going through London

I will be traveling to Oxford, UK this summer.   My flight lands at Heathrow around 12:30pm on a Sunday; it is on British Airways so I believe it will arrive at Terminal 5.   Based on this map and other resources, it appears that one can take a train from Heathrow to Hayes & Harlington station, and then change to a train bound for Oxford, thus avoiding the need to travel into central London and backtrack.   
However, on closer inspection, it looks like the trains from Terminal 5 stop only at Terminals 1,2,3 and then run express to Paddington Station, without stopping at Hays & Harlington (I think this is the so-called Heathrow Express).   So I would need to first transfer to one of the other terminals' rail stations and catch a train from there.   

Is it reasonably convenient to transfer between terminals like this?   Some sources seem to suggest that the best way is to take the Underground to the other terminal.   How easy is it to get from the Underground station to the rail station at the other terminals? 
Would it be better to transfer to Terminals 1,2,3 or to Terminal 4?   
Are there other good options that I am missing? 

I will probably not have an excessive amount of luggage. 

A:

If you just want to get from Heathrow to Oxford - Don't take the train! 
Until they get round to building the western approach spur into Heathrow, taking the train between Heathrow and Oxford is a faff, and it isn't all that cheap either.  You either need to take a train into Paddington, change platforms, then head back out to Oxford (minimum journey time 1:20, likely closer to 1:45), or take a Heathrow Connect train to Hayes and Harlington (change at T123 from T5) then lug your stuff up and over to change platforms then take a slow stopping train. 
Instead, what you should do is take the Oxford Airline bus.  It starts at the central bus station for T1/T2/T3, then goes to Terminal 5, then onto the motorway to Oxford.  The buses are frequent - every 30 minutes for pretty much the whole day, and sometimes even every 20 minutes at peak times.  Depending on traffic, you're looking at between 60 and 80 minutes from Terminal 5 to the centre of Oxford, quicker if you're going to Headington. 
The buses are new, have free wifi, have power sockets, have a toilet, have OK leg room, plenty of luggage space etc. 
The bus stop at T5 for the airline bus is pretty much straight outside of where you come out of baggage claim and customs, just dog-leg round all the people waiting outside baggage claim and head outside.  Stand 10, the dedicated stop for the Oxford Airline bus is pretty much there.  Handy tip - if you have bags, queue up about 10m to the right of the bus stop, and you'll be in the right place to put your bags straight into the luggage storage area! 
(I've tried the train a few times, but via London Paddington and changing at Hayes and Harlington.  Unless you have a very strong reason to need to take the train, such as needing to stop off part way, the coach wins every time!) 

A:

I strongly recommend the "Airline" bus service over the train.  I live in Oxford and fly from Heathrow several times a year: I always use the bus and wouldn't even consider the train. 

It's cheaper: about £30 vs about £40 for a one-month return. 
It's faster: about 75-90 minutes for the bus, vs 100-150 minutes for the train via London. 
It runs 24 hours, though only every two hours in the middle of the night; there are no trains between around 2345 and 0500. ( Included for completeness: not relevant for you since you're arriving at midday.) 
It leaves directly from the Heathrow Central Bus Station (Terminals 1–3) and from Terminal 5 so you avoid having to mess around transferring terminals at Heathrow, which is way harder to do than any other international airport I've ever been to, except Paris Charles de Gaulle.   (Terminal 4 is a pain however you try to travel.) 
It takes you straight into the centre of Oxford, saving you a 10-minute walk from the railway station. 
It's a direct journey versus having to change once or even twice on the train.  IMO, this is a huge win when you've just come off a transatlantic flight. 
It's more comfortable: a long-distance-style coach with reclining seats, footrests and decent seat pitch, rather than a commuter-style train. 
The driver deals with your baggage, vs you having to find space for it somewhere on the aforementioned commuter-style train. 
The bus has free wifi. 

The only disadvantage of the bus that I can think of is that, as far as I recall, they only accept cash.  However, you can buy a ticket online in advance and there are plenty of cash machines at Heathrow anyway.  If you buy a ticket online, it's valid on any bus for 24 hours after the time you choose.  In theory, you only have a seat reserved on the bus you book for; however, I've never been on one of those buses that's been more than half-full so gambling on an unreserved seat is hardly a gamble at all. 
If you're worried about not knowing where you are on the bus, the driver announces the stops and, in any case, if you're going into central Oxford, that's the end of the line. 
If you really do want to take the train, take whichever trains the timetable says are fastest.  That will probably involve going into central London but the lines to Heathrow and Oxford both go from Paddington Station so at least you don't have to navigate London.  If you try to optimize for distance travelled, you'll end up on slower trains that will probably take longer.  Note also that there's only one train an hour from Heathrow to Hayes and Harlingdon, which can leave you standing around for a long time. 

A:

Besides taking a train, you can always use a coach (i.e. bus) to get from Heathrow directly to Oxford.   There are a couple of options.   One choice is Oxford Bus The Airline - it stops at terminal 5 as well as the central bus station and takes about 80-90 minutes to reach Oxford.  The coaches run every 30 minutes.  It'll probably be cheaper than a train, too.