In theory, yes, it's possible to lift and relay existing turf, but there are a few caveats. You will need to cut and lift sections of turf evenly, so that the roots underneath are relatively even in depth on every piece, otherwise it will be very difficult to relay it evenly. Second, once lifted,the turf can be rolled up, but not for longer than 48-72 hours; after that,the grass will start to deteriorate, so it will need to be relaid very quickly, or spread out elsewhere, grass side up, and kept well watered. Third, the shape of the turves you cut should be longer than they are wide and with the majority all the same size to facilitate laying the turves correctly, in straight lines, with the narrower ends in the middle of the side section of an abutting row of turves. This works better in terms of the lawn bonding together than just having, say, square sections all laid like tiles with all the joins meeting all round.
If its really 'thatchy', that's difficult to deal with at this stage; you can dethatch whilst it's still down, but the lawn takes time to recover from that, so unless you're prepared to wait some weeks till it recovers,that's not really a viable option. You won't be able to dethatch shortly after relaying either, you will need to wait some weeks for the lawn to root back in and bond together properly, so realistically, that's autumn at the earliest.
Note that, at this time of year and ongoing throughout the spring and summer, you will need to keep the area well watered after relaying (preferably using a sprinkler) until it has thoroughly rooted in. As you seem to be in the UK, it's probably wise to give the lawn a good soak before attempting to lift the turves, as its been quite dry and warm here in recent weeks, so put the sprinkler on for an hour on each section, then wait a couple of days, cut it to about and inch and a half to two inches in length, and then start lifting.
Hopefully, the turf that was laid was not grown through plastic mesh - if it was, that will be very difficult or impossible to cut and lift.
Lastly, if it is your intention to simply redistribute the soil to create a slight fall away from the house, check what your neighbours' levels are like - you do not want a lower level of lawn at the upper end than that of your neighbours because this will likely cause overflow of excess rainwater from next door's garden/s. You may also find you uncover pipework that isn't far beneath the surface - knowing where the sewage drains and pipes run prior to starting is useful. If your house was a new build, you may come across buried rubble and the like - builders often do this, and any you find is best removed.
