I am regularly waiting for passengers on SQ324 as an airport rep. This flight is late almost every day. Not much, but if it is only 10-15 min late and lands at the infamous Polderbaan (taxi time 15-20 mins to the gate), it is quickly 7:45 - 8:00 AM before you are on your way to passports/luggage. At this time, it is very busy at immigration as many long distance inbound flights arrive at this time. So getting through passportcontrol can take a while, also luggage delivery may take some time. When you exit from customs, you need to go up one level immediately to departure level, BA checks in at row 19 most of the time which is right above your arrival level. You are booked on a BA flight to London City, but at the same time they are handling Heathrow flights, it can be busy at BA check-in. Lately, i have seen only 2 agents at BA check in as there is (like everywhere) a shortage of staff. Please check in online for this flight. You will still stand in line though to drop off your bag. And finally, you indeed have to go again through security and passport control. Plus quite a walk to the gate as BA to City airport usually leaves from the end of the D-pier.
Now to answer your question: is this doable? it will be tight but it is doable. You may need the help of some fellow travelers who let you go ahead of them in the line though. Personally i would look for a later flight to London (Heathrow or City) to have much more ease of mind