You need to contact your local council's social services department, who will put you in touch with the section that supports and advises childminders. They will tell you how long the process takes and whether it's worth applying now.
As part of the registration process, your home will be assessed in terms of safety and will be addressed at subsequent inspections.
The inspection/assessment process will not penalise you just because you live in a flat. As long as it's safe and you can provide suitable care, that's what they'll be looking for. And for what it's worth, my son's (now 25) first childminder was a single parent in a two-bedroomed council flat and I had no worries whatsoever.