That is a good question.
You need a felling licence for things like oak - if you take more that 5 cubic metres per quarter.... rather complex. As you are in a sssi I would get advice and permission before felling them. The oaks may well be protected.
Some people think they should never be felled
some think that thinning them out is a good idea. More room means better crowns on the oaks left behind (better for bugs and birds) but every woodland is different and has different needs.
I suggest you have a variety of experts walk around with you and discuss the matter. All of them will have slightly different opinions - but will know the law, things like needing to check for bats!
It is good to take a lot of different opinions and then make up your mind based on the information given! For example, the guy who wants your trees might want all the lovely straight ones - but if you are thinning you might want to leave the most stable trees to stand for another 100 years and for him to thin the wonky ones that are leaning over precariously.
The price is tricky. If you fell them and extract them yourself you will get 3 or 4 times more than if someone buys them standing. The person buying them standing is taking a bit of a gamble about what the timber will be like inside. They also have to do all the work... and extract them.
The price will also depend on how easy it is to get them out. If your wood has ditches and mud everywhere the price will drop as extraction is difficult.
I suggest you get a few different quotes before you sell them. Also, make sure you are clear on a number of issues
Time of year to fell
Who clears up the brash
If they have a fire, where will they site it and how big it will be
What weather can they extract in? (wet weather and tractors don't mix well)
Do they have insurance and chainsaw ticket?
Can you get recommendations from other people for their work?
If they make a mess, how can you force them to clean it up? (remake tracks etc)
Your felling license also might require you to replant. This will need some thought too.
I hope this doesn't put you off - it is a big decision, take your time with it!
Tracy