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There are two options, both are do-able, and it depends a bit on the arrival gate layout. Many airbridges have two corridors, most people will go downhill towards arrivals, however there is an uphill corridor too, marked international flight connections. There will be someone guarding the entrance to this, and you will therefore need a boarding pass to show the onward connection (and this may lead to some perplexity for obvious reasons). No security check, you are straight back at the gate. The BA lounge is some distance from there, in all probability, but at least the connection is foolproof. These airbridges are used for arrival from European location, but some airbridges only have the downhill corridor, and these airbridges are used for connections from non-sterile locations. BA sometimes ends up here, I suspect TK has first dibs at the dual lane airbridges.

So in this case, follow the main path to arrivals/connections and then you can go back to airside via flight connections, albeit after a security check, and it's still very much do-able. You won't need a visa for Turkey either. Moreover in this stream there is a special area for collecting boarding passes if you don't have one already. BA in IST is handled by Havaş, and they have a desk in this area. Havaş handle a lot of airlines, so that is the sign you need to look out for, rather than BA.