Every month there are different lawn flowers in bloom. This picture is April, with primroses and dandelions, but wood anemone, dog violet, lesser celandine and of course daisies all look great in lawn at this time too.
So better than just one six week lawn, try a patchwork lawn, where one patch is in flower while other areas are mown. over the course of the season the flower patches move around.
You may find your lawn does not have many wildflowers in it. Try raising your own plants from seed you can collect from wildflowers along road verges and wild places. A seed sown in summer should be big enough to plant out by late autumn. Once they are established they should spread.