Just to clarify:
The Belfast Savings Bank (founded 1816) merged with other Savings banks to form the Trustee Savings Bank in Northern Ireland but it did not merge with Lloyds. Unlike the TSB in England & Wales, TSB Bank Northern Ireland was bought by the AIB Bank (Allied Irish Banks p.l.c.) of Dublin, Ireland and merged with its branches in Northern Ireland to become:
First Trust Bank (otherwise AIB Group (UK) p.l.c.)
The headquarters is @ 92, Ann Street, Belfast
PJC (retiree of First Trust Bank)
p.s. I've cut out a few interim steps along the way to simplify!