London UK presumably; if that's the case, the soil there will be very dry currently (I live in London myself). Because there is a central concrete path, I'm guessing this is an older property; decades ago, it used to be traditional for people to bury anything they didn't want in the ground, and that might be why you're finding so much rubbish in the soil.
You really need a hose and sprinkler to soak it all to get it nice and damp, which will make it a lot easier to dig over. Then, using a garden fork, dig the whole thing, removing rubble, plastic, weed roots, bricks, larger stones/pebbles. Using a rotovator won't be particularly helpful at this stage - they don't really go deep enough and you may damage the blades if there's lots of rubble type rubbish in there, not to mention it will chop up weed roots, and that might mean lots more weeds over time. Once you've done that, rake it level and tamp it down, leave it to settle for a week or so, then you'll be able to tell whether you need to buy in some topsoil to raise the level a bit to lay turf evenly. In that time, you may find weed regrowth; that should be removed prior to turfing.
The optimum time for laying turf in the UK is October, but it can be done any time IF you are able to keep it well watered, so a hose and sprinkler will be essential for that too.