I am a householder, but when I looked inside my old iron boiler, when it was taken out:
it is basically an iron box, with flames under the bottom. Water pipe enter top and bottom. On gravity flow, convection takes cold water in at the bottom and hot water rises out at the top. Thus, the water from the HW circuit and the Rads circuit mixes inside this iron box. If you are doing a chemical clean, then yes, you can add the chemicals into the F&E tank after draining enough water out of the drain cock to ensure that on refilling, the water will mix and carry the chemical down into the system. A Chemical DIY clean on an open-vented system like yours is very easy and cheap. I would fully open the gate valve on your cylinder return to get a better flow through during cleaning of that part.
It is possible to just add a pump to the HW circuit, controlled by a Cylinder stat (I did this recently to my old mum's old system, but the pros frowned on the way I had done it). The pump need not be as powerful/fast as the radiators pump.
Usually a fully-pumped system has one pump, and a motorised valve that sends the flow to the cylinder, or to the radiators, or to both. It also receives signals from the cyl stat and the room stat, and sends a signal to the boiler telling it to fire, when required. A 28mm 3-port valve is quite expensive, and the re-piping and wiring take a while if you are not used to it.