You can pay extra at http://www.ba.com in Manage My Booking online to upgrade your class of travel, starting from about fifty pounds in Club Europe (CE). This gives you Galleries/Terraces lounge access.
You can pay for Qantas Club Access.
You earn enough Tier Points to gain a Silver/Sapphire oneworld card which (in the UK) allows BA Terraces/Galleries access on any BA fare. This can be gained for residents of Continental Europe after just ten CE sectors.
You cannot pay to get into a BA Lounge *except* at some (usually non BA owned) outstations which share the lounge with Priority Pass customers, where you can use you Priority Pass to gain access either as part of your membership or for a fee, depending on your Priority Pass membership level.