I've done this recently for 11+ entry and my daughter about to sit the exams in a week or so. We decided last summer to go for a private secondary as the state secondary in our catchment is appalling. The schools we targeted use an internal exam for assessment, issue conditional places and then the Common Entrance Exam as a final step. One of the schools included an interview and written work for assessment also. As far as I gather not all who attended assessment were offered a conditional place.
I spoke to the schools regard their internal exams and was told no prep was needed. The assessment would cover Mathematics, verbal and non-verbal. We did some non-verbal prep - a book I purchased from Amazon!
For the Common Entrance I downloaded the curriculums for each exam subject and compared to the state curriculum (I was surprised at how many gaps there were..). Everyone I spoke to said to get a tutor, and that 6 months was not enough time to prepare. My brother, who is a teacher in a private school, strongly recommended I did not get a tutor, instead that DD work to close the gaps independently and do past papers as practice. We got some revision books from Galore Park and an array of past papers. DD has studied for the last 6 months and done one test a week. DD feels confident and prepared. Her past paper results have increased progressively.
It was a lot of work, and she's given up a lot of personal time but she wanted to do it. For me it was always her choice, she could choose to stop any time ( I've felt immensely bad about how much work she's doing at only 10!).
I'd say that's the biggest take-away, it's possible but they need to want to do it. Else the study & time commitment might just be too much. 13+ is also a lot more subjects at 13+ than 11+.
Good luck!!