Ask Mr. Smarty Plants

If you read the Pruning Guidelines for Prevention on the TexasOakWilt.org page, you will see that most of the infection by the oak wilt fungus (Ceratocystis fagacearum) occurs through the interconnected root systems of the trees.   Since you don't have oak wilt in your neighborhood, it is unlikely that your tree will contract the fungus that way.  It is true that Nitidulid beetles can carry the fungus from an infected tree into wounds on other oak trees.  However, the period that beetles are active is spring—defined by the arborists studying the disease as February 1 through June 30.  The experts do recommend painting any wound with wound dressing or latex paint.  However, the recommendations that I have seen say that for the dressing to be effective, it must be applied immediately.  The sap at the site of the wound attracts the beetles, but if painted immediately the beetles are less likely to arrive in time to land at the site and leave fungus spores.  The wound on the tree is susceptible for about 3 to 4 days if left unpainted, but after that time is not likely to be contaminated.  For more information you can read To Paint or Not to Paint, That is the Question.

Since there is no oak wilt in your neighborhood and the nicking of the root occurred in December, I think it is very unlikely that your oak is going to be infected by oak wilt in the immediate future.  It would be a good idea for you to visit the Caring for Your Oaks page on the TexasOakWilt. org page.