I have wrist issues from previous injuries (thanks open-turns from high school swimming). What I did was i locked my wrists (aka have my hands in a fist and place my knuckles on the floor) so I didn't have to bend them. This works pretty well for burpees (at least for me). I've also seen people "elevate" their burpees so that their upper body is on a chair or block so that they can get rid of the pressure on their wrists, this didn't work well for me, but feel free to try it.
Until I was ready, I would do push-ups on my knees (i added the weight vest to get more resistance as i got stronger) and then worked my way up to be able to hold my body weight (not the vest or anything else) with my wrists in a normal push up position. When my wrists got tired, I would shake them out, go to my knees and resume push-ups.
For planks, I would go down to my elbows and alternate going on to my hands to push myself after getting strong enough.
Hope this helps!