IIRC (and I may be off, bear with me I'm tired and have been drinking rye and coke) Dowding was asked of Churchill the status of his reserves to which he replied "we have none"
You do remember wrongly.
Churchill visited Uxbridge, not Bentley Priory, on the 16th. IOW, he visited the HQ of #11 Group, not the HQ of Fighter Command. He talked with Park, not with Dowding. There he saw that in the late afternoon, at a time when new hostile plots were moved on the map, #11 Group had no ready reserve.
And he was deeply impressed. He later told his aide, Gen. Ismay, that he had "never been so moved". Ismay himself recorded that seeing that situation, he felt "sick with fear".
You may have conflated this with Churchill's account of an answer by Gamelin, who, according to Churchill, did say "aucune", when asked, a few weeks before, about _ground troops_ reserves.
But neither Ismay nor Churchill understood the whole situation, and plenty of other people have misrepresented it based on their feelings.
For starters, while #11 Group was certainly the front line, it did not control the fighter Squadrons of the other three Groups, so the remark about "no reserves" never applied to Fighter Command as a whole. Units could be moved to #11 Group from the others; later in the battle, #12 Group's Squadrons repeatedly took off in order to cover #11 Group's bases while the latter's units were engaged.
There's more. Park had no _ready_ reserves _at that moment_. Neither Ismay nor Churchill could know how many Squadrons were being refueled and rearmed in that very minute, and be ready to take off again for a second or third sortie in a few minutes more. Park could and did know that.
Park certainly was operating with a narrow margin on that day. Considering how he managed the battle and how seldom was he taken by surprise, we can say that it's highly unlikely it would have happened on that day, at the end of the afternoon by the way.
It is not entirely outlandish to suppose that Park may have not explained Churchill that the situation was not as dire as Churchill seemed to believe it was. After all, what do all commanders want for their commands? More troops. Reinforcements.
Indeed, the air operations on that day were not among the most significant in the battle. The Germans managed to score a couple of good hits on air bases – but both had nothing to do with a lack of reserves and indeed did not take place at the time we're discussing. OTOH, on the day before the Germans had carried out a great effort – and the loss rate had been 2.3:1 in favor of the RAF, and two days later it would be 2:1 in favor of the RAF.