I would call them and follow up to make sure everything is ok and maybe throw in the 'have had other parents contact me looking for an interview so if you want to move forward and secure the space I need the deposit by X date'?

I never consider a space 'filled' until the deposit is in my hand and contracts are signed ... and that comes out of my mouth automatically now in interviews because when I was 'new' to daycare administration I did the same things and 'assumed' that people said what they MEANT but in reality they were just stringing along to ensure that they could finish interviewing at other centres and so forth by telling you they wanted the spot - than when they were sure to have their 1st choice they would call the centre and say they'd changed their mind and I would have to go back to booking more interviews for spaces.