Tenant applicant slow to response

[Jeremy Szymankowski]

My current tenant has been with me for a year, and their lease is expiring at the end of the month. 

I listed my condo for rent online, but in the process, my current tenant said they would like their mom to move in to retain the residency and keep their furniture inside the condo. So, I gave the mother an application to complete, and it took her nearly a week to complete it and get it back to me. During this time, I had received many inquiries, and had another woman fill out an application in which she completed within the day. This woman is prompt, communicative, and seemed very excited for the condo. I would certainly feel comfortable moving forward with her in the application if I weren't working with the current tenant though. 

So what do I do then? The current tenant paid on time, and took care of the property, and so I trust that if their mother takes over, things will still remain orderly. She has good credit, and no debt, but she has just been very slow at filling out the papers and getting back to me. Now I am waiting for some paystubs and bank statements. After that, we'll have to do the final lease paperwork, along with the security deposit and I don't want each of these to take a week to receive.

Is there any courteous approach in telling her that I need these papers or I will have to move on to an alternative candidate? I would just like her to be a little more prompt throughout all this, though I feel this is out of my control. Should I just go with the person who is more prompt and communicative? 

Note: This is my first condo I am land lording for.