Same day service is definitely the way to go. I had to change my JAA ATPL for a EASA ATPL a couple of months ago and I arrived LGW at 0630. I was number 3 in the queue and sorted by 1100. Number 1 in the queue had driven from Aberdeen and the second guy had flown in from Europe the night before. The guys and gals at the same day service desk were their usual helpful and friendly selves but even they intimated that there was a big problem brewing.
I think the thing to remember, and it isn't easy, is that CAA have had EASA dumped on them just as much as we have and it really isn't their fault that the system can't cope. We need to look several rungs higher up the food chain for that.