What about using a SortedDictionary<string, uint> to sort using a custom comparer?

Proposed code:

public static ReadOnlyDictionary<string, uint> GetWordCountDictionary(string stringToCount)
{
    Dictionary<string, uint> wordDictionary = new Dictionary<string, uint>();

    //Rest of the method here that is not relevant

    // Custom comparer can be created to order differently, instead of `StringComparer.InvariantCultureIgnoreCase`
    var sortedDictionary = new SortedDictionary<string, uint>(wordDictionary, StringComparer.InvariantCultureIgnoreCase); 

    var result = new ReadOnlyDictionary<string, uint>(sortedDictionary);

    return result;
}