Hi, we did assessments from November to Jan this year at a number of London schools - not the ones you mention though. We didn't tutor, but DD goes to a good kindergarten which I'm sure helped. Assessments varied from school to school, but could involve listening to a storybook and answering questions, drawing a picture of your family, writing your name, recognising and grouping similar objects or shapes, replicating models made of duplo/Lego blocks, and group play. I've heard of schools also doing puzzles, lacing, threading beads but as far as DD was willing to tell me, none of her assessments included these skills. They also didn't do any sport or physical activity (that I know of). Reading and writing was not assessed, other than the child recognising or writing their own name.
In terms of what you can do :
-I know that some people register their children to schools they are not interested in, just so the child gets a "practice" assessment before their "real" ones, to get comfortable with the setting.
-There are lots of good apps that can also help your DD with maths and reasoning. If you do a MN search you will find loads of suggestions.
-If you want to improve her vocabulary, I would recommend books such as the Hairy McLary ones, Katie and the Sunflowers (and others in that collection), Julia Donaldson's, Harry and the Dinosaurs and generally just lots and lots of reading.
For those who say "don't put your child through that", my DD had a ball each time. I told her they were "special playgroups" and she had/has no idea she was being assessed. She is not the most outgoing child, although she isn't shy, but the schools obviously have tons of experience at putting children at ease. They are assessed with children of the same age so you don't need to worry about her being young - she will be compared to her age peers. The highlight for DD was being given a biscuit at the end of each session. It was certainly not an ordeal!
Some people on MN seem to think that only the most outgoing children perform well at assessments. That was not my experience. I also think that schools are probably looking to get a mix of personalities - who would want to teach a group of 20 alpha kids? They need some compliant / cerebral kids to counterbalance the very energetic/ dominant ones.
Good luck, I walked in your shoes and I remember the stress! It will all be fine in the end.