The child could likely be called George, and then one or two middle names from William, Augustus, Frederick, Leopold, Francis.
Leopold probably stays in Great Britain, and is involved in the upbringing of his son. As Leopold was a military man, it seems likely to me that he will give his son a militaristic education. All things considered, he seemed to be a loving wife to Charlotte, so I can see him being quite a good father, as far as the rest of the Hanoverians were concerned as fathers to sons. The main issue is that George IV will also want to be involved in his grandson and heir's education, especially as the child is to be head of the Church of England, and so will provide religious instruction. Whether this causes him to butt heads with Leopold or not, I am unsure. But Leopold being granted the title His Royal Highness in the UK will probably still happen, and maybe he would be granted a ducal title in the UK. It had been proposed that Leopold become Duke of Kendal, so that could well happen here.
Assuming his father dies on schedule and the boy is 13, then there will be a regency. The main candidates would be Leopold, and George IV's brothers, William, Ernest, Augustus, and Adolphus, with preference probably for Ernest. The regency will be provided until the child reaches 18, but the child may well be allowed to take on royal duties at 16.
The other factor is that Hanover cannot pass to Charlotte's son through herself, so it goes to William, who becomes King of Hanover as in OTL.