Both options `PlotTheme -> None` and `PlotTheme -> "Classic"` don't give the expected coloring for points. However, you can set their color explicitly using `PlotStyle -> Directive[PointSize[0.03], ColorData[1][1]]`:

    ListPointPlot3D[
      Partition[Flatten[Table[l u1 + p u2 + q u3, {l, -3, 3}, {p, -3, 3}, {q, -3, 3}]], 3], 
      PlotRange -> {{-1, 1}, {-1, 1}, {-1, 1}}, 
      PlotStyle -> Directive[PointSize[0.03], ColorData[1][1]]]
>![LPP3D][1]


  [1]: https://i.sstatic.net/khHaY.png