This is actually implemented (as well as a host of other features) in the latest version of Sage. This is the culmination of 2 and a half years of hard work by Stefan van Zwam and Rudi Pendavingh, together with help from Michael Welsh and Gordon Royle. See this page from the Matroid Union Blog to get started. For your particular question, it is easy to construct $U_{2,4}$ via the Sage command
Sage: N = matroids.Uniform(2,4)
To test if an input matroid M has an N-minor you can run the Sage command
Sage: M.has_minor(N)
As you can see, Stefan and Rudi have worked hard to make the syntax easy to understand. Of course this is a very generic approach, so I am not sure how optimal it will be. Feel free to contact Stefan if you have any questions, or (better yet) want to develop for the package (Sage is open-source).