Get a koala?
Cut the branches into logs, keep them under cover for a year then split and burn them (if bone dry) on a barbecue and cook over the embers (I have seen this done in Zimbabwe and the eucalyptus wood cooks and flavours the meat nicely).
Not sure about the twigs and branches, compost?
The council will collect if it is bagged ttp://www.haringey.gov.uk/environment-and-transport/refuse-and-recycling/recycling/garden-waste-collection to order the bags; however if there is a large amount I have also used green plastic bags and they have still collected the green waste.
This begs the question of why the person/people who removed them didn't take them away, a normal part of any deal. I hope they've not damaged the tree by trying to do it themselves as tree surgery is a really skilled job. It would cost, of course, but they could get a tree surgeon to bring their grinder round.