Are you flying Y ? this is the availability on the AY 10.20 flight - on the BA code it is full in Y
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Departing LHR on 05/18/17 12:00 AM for HEL
Flying OneWorld
Point of Sale: UK
Frequency
Flight Stops Depart Arrive Aircraft Reliability Available Classes
0 Connections
AY 832 0 LHR HEL 359 Daily J9 C9 D8 I0 Y9 B0 H0 K0 M0 P0 T0 L0 V0 S0 N0 Q0 O0 Z0 R0 G0 W0 X0 E0
05/18/17 10:20 AM 05/18/17 3:15 PM 80% / 13m
0 Connections
BA (AY) 6032 0 LHR HEL 359 Su,M,Th J9 C6 D6 R6 I6 Y0 B0 H0 K0 M0 L0 V0 N0 S0 O0 Q0
05/18/17 10:20 AM 05/18/17 3:15 PM 79% / 13m
But you'd only gain 10mins even if you make the change. Arguably you are introducing more risk - your aircraft is in London, so I suspect they're having a crew issue and the flight is delayed awaiting minimum rest for the crew that arrived last night so the time is locked in now, the A350 could be delayed on it's way in for any number of reasons
From what I can see your only hope is either
a) go to LHR early and try get a reroute on another carrier - e.g. SAS via Copenhagen
b) proceed to Helsinki and hope they can get you onto the Tallinn flight at 4.25pm (currently full) - which will enable you to return back on that same aircraft
c) delete the TLL segments from both tickets and turn back at HEL - not convinced you'd have much success at this but could be worth a go
otherwise you'll need to change your nested ticket - they've no obligation to but you might have some success with this at the airport for free, though you'll need to overnight in Tallinn or Helsinki
Good luck!