I don't know a lot about Whittlebury I'm afraid... The White Horse in Silverstone Village is not too bad. Although it's a good 20-30 minute walk from the main gate at Silverstone. I'm not sure how long it would take from the Whittlebury Hall campsite, but I'd say a good 20 mins on top of that. Did you walk from Whittlebury last time?
I stayed at the Saracens Head years ago and it was always clean with good food/beer. I hope it's the same now. It's definitely not a walk away from Silverstone/Whittlebury though.
A word of warning as well... Any taxis called out to Silverstone village or nearby hotels usually charge double fare/call out from Towcester. Stu and I were due to stay at the Premier Inn, next to the Green Man Inn on the A43 but the taxis were charging over £20. That hotel is just a couple of miles to the circuit, but a walk along a dual carriageway. We found a B&B in the village and walked in.