Hyde Park is really overrated and will be horribly busy, I would avoid!
If I were you, I would go to Southbank (really easy to get to from Waterloo, so a good starting point!) and look at the Christmas market, have a wander along as there's loads to see around there.
Then walk across Waterloo bridge which is my fave London bridge - you can see Parliament and Westminster Abbey, the Shard, the Eye etc. On a clear winter's day the view is amazing! Once you're on the other side of the river you're right by Somerset House, which I LOVE at Christmas. You walk through the arch and there's a massive Christmas tree and the ice rink. The decorations are so beautiful! You can either go skating or just watch others doing it for a little while.
Next, you're a five minute walk from Covent Garden, which would be a perfect place to spend the rest of day - you can shop, look at the lights, walk around the main market square or go up to Seven Dials which has more cool, independent shops and is lovely at this time of year. There are lots of great cafes and restaurants for lunch/dinner (plus stalls selling mulled drinks in CG!) and then you're right by all of the main theatres if you did want to take in a show. I would really recommend Matilda which I saw recently and loved.
I really think that bit of London is so lovely at this time of the year without the madness of Hyde Park. Covent Garden will be busy, but it never feels as bad as HP or Oxford Street.
Hope that helps, OP!