The cargo tunnel which is an airside tunnel connecting the central terminal area and T4/cargo is currently closed for refurbishment. This tunnel is used by the airside buses which go to/from T4 to T2/3/5. I am afraid I can't find out when the work is going to be completed by, so if anyone knows that would be useful to know.

The impact to the closure is that the only airside buses running to/from T4 go to T2 - there are no direct airside buses from T4 to T5 or T3. The airside buses take a long and slow overground route now between T4 and T2 which goes around the eastern edge of the southerly runway. If you need to get between T5/3 and T4 you will need to get an airside bus to T2 and change.

Until the cargo tunnel reopens if you have a UK/EU/EEA passport I would strongly recommend taking one of the landside options if going between T5/3 and T4 - so TfL buses, tube, or HEX.