Visa4UK Frequently Asked Questions addresses your concerns, as you can immediately book an new date, the Update Appointment option.
If your appointment is made through Visa4UK and you have also paid online, then you can:
1. Click the View Appointment button
2. Click either the Update Appointment or Cancel Appointment.
Why is my preferred Appointment Date unavailable (greyed out)?
At some locations a restriction has been set which prevents applicants from choosing a date within a period of time immediately after the date when the applicant is attempting to book the appointment. These restriction periods may vary i.e. some applicants may find that the next day is unavailable, others may find that more than one day is unavailable. In addition to these restrictions other days and times may be unavailable for various reasons e.g. public holidays.
No mention is made of a time limit between between fee payment and appointment booking; at some centres, it can be months before slots become available. Should you have an undue delay, you may wish to contact the UK Visas and Immigration Help Centre, a fee based service.