If you fly LHR-ORD you can either fly ORD-ATW (Appleton, which is KOSH's closest commercial airfield) or hire a car and drive up from Chicago. I have flown into MSP but the drive is much longer. You can either camp on the airfield, rent an apartment nearby or book a hotel. There is a Hilton on the airfield but it'll be fully booked already for EAA 2018.
Air Venture, as it's called, runs from Sunday to Sunday however I would only recommend attending for a few days (perhaps the Thursday, Friday and Saturday). By the final Sunday most exhibitors have packed up and left and a large number of aircraft will have already departed, the airfield is left feeling a little deserted and the excitement/buzz has gone.
I would highly recommend being there for the Saturday - there is an awesome BBQ down at the Seaplane Base followed by a spectacular night time airshow in the evening.
I would personally recommend looking into camping on the airfield or as close to it as possible. There are excellent shower facilities around the show ground which are free to use, the whole thing is expertly organised and you'll have a great time (assuming the weather plays ball).