Presuming this is for your dc, it will depend a bit on their age.
In my authority, the EPs usually arrange one appointment with the parents and one with the school / Nursery. Sometimes, for convenience the parents come to the school / Nursery and this happens on the same day, particularly if both parents and Nursery are 'on the same page'. Sometimes the EP will go to the child's home - partly because some parents find this more comfortable, and sometimes so they can see the child in a different environment.
Most EPs like to meet the family first, but not all, and again, it depends on diary pressures.
At school / Nursery they will talk to staff - usually both SENCo and either class teacher / Key Worker, or sometimes whoever works closely with the child as well - LSA / Room Leader maybe. They will also observe the child in the classroom, and (if the child is older) they are likely to work 1:1 with the child, both to give the child the opportunity to be involved and express what they want, but also sometimes to do specific assessments - maybe understanding of language, or something. Pre-school they don't tend to work 1:1 with them.
At home, they talk with the family about the whole history, about things like sleep and eating, and about what the parents concerns are, and what they feel the child might need, and (this is really variable, dependent on child) the hopes and aspirations as well as the concerns.
I have only worked in one LA so don't know how much this varies in different parts of the Country.
Oh, re waiting, again, it depends. I work with pre-school children... November referrals are meeting their EPs about now. Pre-schoolers are only referred after getting support elsewhere and when they need to apply for an EHC here. Schools however, tend to have waiting lists as they are allocated 'hours' and another child might take priority. Systems definitely vary in different LAs, some of which don't use EPs pre school, for example.