Hello Annie
Sorry to hear about your nettles – clearing ground is undoubtedly harder than keeping it clean so I understand your frustration. Nettles are actually a sign of fertility so you have some good soil waiting for you and now the nettles are, presumably, dead (!) it should go more easily.
Just recapping, I suspect they grew through the cardboard mulch because it may not have been overlapped enough (leaves coming up the cracks) – also it may have decayed enough after two or three months for vigorous shoots to grow through. Cardboard needs replacing after ten to twelve weeks because it does rot surprisingly quickly, especially where something is weighting it down.
I would do another cardboard layer and continue with erecting the tunnel. Soil will be full of dead or dying nettle roots and probably other weeds. If the tunnel is in place before winter, you could plant some winter salads in early October, otherwise it may be best to leave soil covered for now and become really clean. Also slug numbers may be high at present as there are often a lot after ground has recently been cleared of their food supply; then their population reduces after a few months as there is less to eat. Maybe take all mulches up in January or February and clear the ground of any remaining weeds over a few weeks, before sowing and planting. If you are growing tomatoes they do not go in until early May so you have plenty of time to prepare the soil – which will be a bigger job in this first instance, and much less work thereafter.
So be prepared to really give it a go in these coming months, to get the soil really clean and also fertile with the mulch of manure on top.