Kept my horses in the middle of Burnham Beeches some years ago. The roads were pretty quiet but you aren't allowed to ride in the woods themselves. Wasn't overimpressed with the hacking but I have been in places that were worse.
Yes you can hack on the tarmac. There is a bridlepath in the big field by Snowball and also across the road. It might be worth popping into Snowball to ask the liveries about the hacking.
I know Burnham like tha back of my hand grew up there & only just recently moved.
You can only hack on the tarmac through the beeches & they have recently installed very un-user freindly gates at both main entrances so you have to dismount to get them open.
There are some great bridleways around the snowball farm area but there is a really dodgy road to negotiate to get to them.
If you are stabled up towards Jennings then there is nothing except roads.
Taplow is better, some roadwork but a lovely hour or so ride through huntswood once you get to it