Well, that was fun. As advised above by craigthemif and Yilanes, security check was required (which I should have remembered, I recognised it immediately when I got there, though it's been a couple of years since I did a T4S>T4 connection, in defence of my defective memory )

My 60 min connection was turned into a 33 min connection by the BA flight LHR-MAD being delayed. Cue a sprint through T4S, lifts, Immigration, train, security, another lift, another sprint to far end of K gates...I made the flight, was second last to board (the last was another unfortunate soul from my BA flight). Did it in 22 mins from doors open at T4S to reaching the K gate at T4.

Good thing I was one of the first off the LHR-MAD flight, and am familiar enough with transit/immigration to beat the crowd. Thanks to the BA CSM for checking on arrival and connection times in flight, and IB staff being at arrival gate with priority connection wallets. I didn't need it in the end as security was empty, but made me feel better. Also got email/text from IB confirming the gate, which was handy.

I would say that an hour connection would have been pretty comfortable for me, and the 1h20 that the OP has would be plenty (I might have even dodged into the Dali lounge for a few mins with that one).