I found some information on the net that might help you -

"Growing early vegetables such as spring cabbage, winter lettuce and broad beans (Aquadulce) is one solution. Sow in early autumn so they are well established by early spring.

Try growing some shade tolerant vegetables (shade for more than half a day) like beetroot, 'Bolthardy', calabrese, kale, kohl rabi, lettuce - little gem. Also radish, spinach and herbs - chives, mint and parsley are all shade tolerant.

Fruit such as raspberries, blackcurrants, gooseberry and rhubarb are plants of the woodland edge and will thrive in this environment.

You could try other strategies to improve the growing conditions such as increasing the light levels by careful pruning of the tree and raising the canopy.

Improve the soil too as tree roots remove a lot nutrients and water. Work in plenty of well rotted organic matter at a rate of one wheelbarrow/5sq m.
Starting seeds in modules will get them off to an early start with an established root system."

Thank you for your answer, we can't do anything about the oak tree as they have a preservation order. My husband is busy digging again trying to get the old roots out, I am going to put in broad beans and beetroot see how they fare

I hope you manage to get a crop. :-)