Totally agree with what previous posters have written, especially Mardee and Nigel.
Since your trip will be in January, my advice would be different from what I would tell you if your trip would be in June.
The weather may be very cold in January or it may be clear skies and sunny....no one knows what the weather will be this far ahead of time.
We have been to Stratford-Upon-Avon only in warm weather and enjoyed spending several days there walking around. Yes, seeing the Shakespeare sights and two plays. However, we used it as a "break" in the middle of our trip to England one time. We had been on the go pretty hard for a couple of weeks, and chose to stay in SUA for several nights.
We enjoyed the good restaurants and pubs there and just took some time to relax. We watched the narrowboats travel on the canal beside our hotel and helped one older couple with the lock right beside our hotel.
My husband visited a bicycle shop there and the Pashley factory (bicycles built in Stratford-Upon-Avon).
So I would say it's a good place to see the Shakespeare sights, see a play and spend awhile in a nice little town by the river. However....
If all you're going to do is jump off the train, walk through the center of town and jump back on the train, I'd say don't bother. Just stick with Oxford.
About Simon's comment: "Both Oxford and Stratford-upon-Avon are easy day trips from London."
This is true, but I'd want more time in Oxford....several nights....and I'd skip SUA totally before I'd do it as a day trip from London.....purely personal preference.
You might watch a few YouTube videos of SUA before you decide whether to go there or not.
There are several videos of a walk around the town that will give you a preview.