The scores for Adams' (as well as other schools in the Walsall consortium) are calculated completely differently to the Birmingham grammars and, as I understand it, you could very roughly compare the two by dividing the Adams' score by 3 and the Birmingham schools' by 2 and comparing the results, which actually means that it is a lot easier to get into Adams'! I think it makes sense as there aren't as many children as in Brum, thus less competition. I hope that this is reassuring for you.
The test takes place on July 1, and the school writes to the parents in mid-September with the standardised scores obtained by their children and the details of the lowest score offered the previous year on the national offer day (I think). The letter also either strongly encourages to apply for a place by putting it on the local authority form in October or not, but this letter does not guarantee a place in itself. Of course, for example, if your son scores 360 with the last in the year before scoring 314 (for 2014/15), you could be pretty confident of success, but if the score is borderline, you would have to wait until the offer day or later as the scores can go up as well as down. They went up this year. I am not sure what the lowest score was on the offer day, only know that we had an offer at 323, and another family did not at 319, so it could be anywhere between 320 and 322. However, do not despair if your score is close, because if another child with higher scores than yours decides not to take up his place, the child with the highest score who was turned down gets an offer instead, and so on, so it is possible for 319 to get an offer still.
I hope this clarifies the system somewhat.
Regarding the preparation, we did not have a tutor, just practiced at home, not very vigorously. I do not think that my DS would have had a chance to get into GS in Birmingham, it appears to be a lot more competitive.