Would it be possible for your to visit separately from your Mom, so that you have more control over the length of the visit? Or even to join your Mom for part of the visit but not all of it? Is there are reason why, when you statt to get bored or feel too crowded, you couldn't leave? Either to retun home.
It does seem as through you have three separate issues:
1 . Your cousin using the computer when you want to
I think this is unreasonable. It is his home, so he takes precedence. Added to which, you feel it is OK for you to be using a computer during a visit, as a guest, and for your aunt to be using her kindle, as a host - so it is extremely unreasonable to try to stop the 10 year old from doing the same. And while he may be rude in breathing down your neck, and not sharing politely on the occasions when you do use his grandma's PC, it's not your job to correct his manners. At most, you can set a good example by your own actions. And if you know that you will get bored and need a distraction, then it is your responsibility to supply that for yourself, whether it is a smartphone, your own laptop or tablet, a set of 'worry beads' or whatever else works for you.
2. The room being overcrowded and with inadequate seating for the number of people
This is a legitimate issue, but it is not the child's fault, not is kicking him out of the room the solution. You can, as suggested, raise the issue with your aunt and ask about whether you can move to a different room, or bring in extra seats. If you can't have that conversation with her, or if there simply are no other suitable spaces, then *you* need to make changes to how you visit - the easiest and most obvious options would be to visit when your cousin will be at school or to visit separately from our Mom, both of which result in there being fewer people in the room t any given time.
3. You feeling that not all conversations are suitable for a 10 year old.
This depends a little on what the conversation is, but ultimately if you are not comfortable discussing something in front of him, don't discuss it. Change the subject and if necessary (for instance, if the discussion turned to something persona bout you, such as your own diagnosis or treatment, you can say "This is quite private, I am not comfortable discussing it in front of N" If it is a subject which you feel is not age appropriate but which is not personal to you then you can chose not to join in with the conversation but you don't get to dictate to your aunt or Mom what is or isn't appropriate in front of the child. If there are things which you want to discuss with your aunt you have the option of visting at a time when your cousin will be in school, or phoning your aunt
You've mentioned that your cousin is potentially lonely and doesn't get much attention from his parents. Another option might be for you to engage directly with him more. Have you considered, at the point in the visit where you start to get bored, speaking to him directly and suggesting that the two of you do something together, such as playing games on his laptop, popping out of the house to get some fresh air / pop to the store for drinks / whatever. It would have the benefit of getting you out of the stressful, overcrowded situation for a bit, alieviates the boredom, and it is possible that if he knows he will get some positive, one-to-one attention from you during visit that he will then more more amenable to a request that he let you use the PC, or to a suggestion that he watch TV or play elsewhere so that he adults can have a chat, at other times during the visit. If there are things you personally want to talk about privately with your aunt you might also be able to ask you mom, in advance, whether she would be willing to take child out for a short while at some point in the visit to give you a chance to have some one-to-one time with your aunt.