I used to be an ed psych in my pre-children days. Please do NOT worry about the ed psych visiting.
The ed psych will probably spend some time observing your dd, some time doing activities with her, lots of time talking to you and also talking to any teachers or helpers that work with your dd. If she visits you at home, she may well try to see your dd in her nursery at a later date. She will know that very young children are very inconsistent and while they may behave one way in one setting, they might behave completely differently in another. For this reason, she will base her reports not only on what she sees but also on what others who know your dd well have to say. And of course, you are the expert on your child, not the ed psych - you've known her for the last 20 months, as opposed to an hour or so!
The ed psych will not get to decide where your dd goes to school. It sounds like her role at this point is to write a report which accurately describes your dd's skills (e.g. her language skills, physical skills, cognitive development, social development etc). This report will be considered alongside other reports (e.g. medical reports, information from parents, info from the school, speech therapy reports etc) and a panel within the local authority will decide which might be the most suitable school for your dd. However, where I used to work, parents wishes were generally taken very seriously, and the Local Authority would try very hard to get the child into the school that the parents wanted. However, obviously sometimes parents did not get the school they wanted and then were very upset.
If I were you, I would not worry too much at this stage about whether to go for "special school" or "mainstream school", but instead look at the individual schools concerned. Some special schools are brilliant, but others are not. Some mainstream schools are fantastic at meeting the needs of children with SEN, others are terrible. I would suggest going to look at the schools, talking to staff and also talking to parents whenever possible.