If an AUP is available I thought there were relatively standard prices, so around $700-$1,000 on the west coast routes. Looking at the seat map there is only 1 seat unallocated in CW. If it was me, and I really wanted that AUP, I'd be getting to the ticket counter the second it opened. If it helps, here're the seat maps for your flight:
Results from https://www.ExpertFlyer.comSeat Map Search:
Departing SJC on 23/03/18 for LHR
Flying BA flight 278 in Business/Club

A B D E F J K
6 - - - - - - -
7 - - - - - - -
A B D E F J K


A B D E F J K
10 W - X - - - X - W
11 W - - - + - - - W
12 W - - - X - - - W
13 W - - - X - - - W
A B D E F J K

Flying BA flight 278 in Premium Economy

A B D E F J K
16 W - - - - - - X W
17 W - - - - - - - W
18 W - - - + - - - W
19 W - - X - - - - W
20 W - - - X X - - W
21 W - - L L L - - W
A B D E F J K




Seats
Premium Only P
Available +
Handicap-Accessible H
Paid & Premium #
Occupied -
Blocked X
Lavatory L
Paid $

Location
Exit Row E
Wing W
Upper Deck U