Realistically you are talking £2K for this trip. I spent a lot on getting home at the end of the trail and some cash could be saved here.

Getting There

Return flights Newcastle - Heathrow - San Francisco - British Airways£990.11 
Luckily I was able to stay with a friend who fed me and took me to Yosemite for the start of my trip - so there were no further costs incurred on the way to the Trailhead. 

En Route

Vermillion Valley Resort - Tralier rental plus meals plus some supplies$218.00
Reds Meadow Resort - Meal$25.00
Muir Trail Ranch - Fee for picking up food drop$55.00

Getting Back

Accommodation - Lone Pine$70.00

Bus Lone Pine to Bishop

Bus Bishop to Mammoth Lakes$7.00 
Private Shuttle to Reno Airport$225.00 
Flight Reno to San Francisco$284.70
Admin Reno to SF$30.00 
Baggage charge Reno to SF$25.00 
Accommodation Reno Airport$113.34
Accommodation SF$140.00
BA Fees to change flight$512.00

Specialist Gear

Maps - JMT Mappack from Stanford Maps£18.75 
Bear Vault - bought from a US retailer£72.40 

Grand Total

Total ($ converted to £ at 1.6)£1964.88
Total above excludes meals while travelling in the US and food for the trail.