JOSM: How do I select a way for editing that is part of multiple relations

To change the geometry of a relation you have to change its members. Because you don't want to change the geometry of all relations you will have to create a new way. The steps are as follows:

• Remove way 140856543 from the wood relation you want to change. You might have to split it first if the first and last nodes don't match exactly the start and end point of your intended geometry changes.
• Draw a new way representing the section of the geometry that differs from the existing relations.
• Add the new way to the wood relation.
• Make sure the wood relation is closed again. JOSM's relation editor has a nice sort function where you can check whether the relation is closed.
• Check whether the other relations you touched are still correct.

I've covered something similar to this as part of the Irish Townland mapping project. This section of the tutorial video should show you what to do

https://www.youtube.com/watch?v=WG6Po8SmweE&feature=youtu.be&t=16m55s

I don't know if you are overly familiar with relations so if I am being overly basic in my instruction please excuse me but scai has outlined the actions required above in his post so I will just expand on his instruction. The instruction is applicable anytime you wish to remove or add something from a relation, regardless if there is 1 or 7 relations sharing the same way.

First thing to note, you don't work on just one relation when you are doing these steps until you get to point # 4 below. Your actions affect all relations that share the way you are working on. This is not a reason for you not to proceed, just something you need to be aware of.

I am also assuming your question relates to adding or removing ways to existing relations to allow you to add/remove bits to the forest.

1. In my video, at 19:20 you will see I select a node and split the way and then select another node and split again. This is to allow me to select the section I want to remove and only that section. Do this with your relation when you have a beginning and end node for the bit you want to modify. This isolates the section you want to remove/extend/modify
2. Next, at one of the points at which you split your relation, you want to draw a new way which reflects where you want your relation to be. The way should start where you did one of your splits and end where you did your other split so that once you edit the relation members you will still have a closed loop
3. Now select the section you split earlier
4. In the relation window, select the relation for your wood/forest and select edit
5. In the window that opens up, the way you split earlier will be highlighted, click the trashcan to remove it from your wood/forest relation
6. Now, without closing the relation editor, in the background, click on the new way you drew, it will show up on the right hand side of the relations editor, click one of the pink/blue buttons to add it to your relation
7. Now click the A>Z button. Does your relation show a loop? If yes, happy days, save your change and upload, if not let me know

Note, the video will go through the adding/deleting bits out of the relation from 19:00 onwards just incase you need something else to reference against