Lawns and wildflower meadows have different requirements. Most grasses are thugs of the plant world, and will out-compete most things if nutrient levels are high. Most of the wildflowers we enjoy aren't very good at competing, so seek out areas where no-one else wants to grow, eg lower nutrient areas (or shade, dry soils, high pH areas etc).
So main thing you will be trying to do is reduce your nutrient level:
a) by planting the hemi-parasitic yellow rattle as others have suggested
b) by cutting in late July when the grasses have as much nutrient as possible in the above ground growth and aren't yet returning nutrient to their roots of the soil, and taking the cuttings away so they can't rot back into the soil.
This may mean cutting before all the flowers have seeded, which is why most grassland wild flowers are perennials and not annuals dependent on re-seeding each year.
c) the gold standard - remove the fertile top few inches of soil entirely, and start off by seeding a mixture of grass and flowers on the less fertile soil below.
If you're not going to start by removing the turf and top soil, then you'll do better by planting plug plants. Seedlings will find it hard to get established in the thick grass. Yellow rattle is the exception - it's an annual and you'll need to sow it direct, but make sure you get it right down under the 'thatch" and in contact with the soil.
Be careful in your choice of seeds. Most "wildflower" seed mixes are annual flowers which normally grow in arable areas - they need bare soil to establish, so are better in a bed that you dig over every year. You need a seed mix designed for a perennial grass meadow.
Good practice for a wildflower rich hay meadow is to allow stock to graze until end May, then close it up for the flowers and grasses to grow, harvest the hay after mid-July, then allow animals in to graze the "aftermath". This is what you'll mimic with your mowing regime.
However - you may want spring flowers (I have wild daffodils, snakes head fritillary, primroses, cowslips, and non-native species crocus and tiny tulips) - this means you can't "graze" the lawn in spring, so it provides more competition for the summer perennials, and also for the yellow rattle, which doesn't like being shaded by tall grass.
And you may want later flowers, like sanguisorba, scabious, knapweed. My experience is that you'll get away with mowing up to late August rather than July - not ideal, and you may want to reduce the height of the grass, or cut it selectively where it's been flattened by wind, rain or cats. But if you leave your mowing until Oct your meadow definitely will decline.
Finally, as someone said - do mow a strip round the edge - it's amazing how much tidier this makes it look.
Your front garden sounds lovely, by the way.