What sort of wildlife/nature?
You may find that the lens is a lot more important than the camera in most circumstances.
Update. HD is video not photos.
You have 3 options.
Relatively Cheap..... A bridge camera with a wide zoom range, macro/close focus capability
More expensive... a DSLR camera with kit lens and extension tubes for macro
Even more expensive... a DSLR camera with specific macro lens for close focus and telephoto lens for other animals.
How much do you want to spend?
To illustrate. The first linked photo was taken with an EOS600D/ T3i, 100mm f2.8 macro lens and ring flash. The camera isn't available anymore but the T5i is $650. The lens is $550 at B&H and the flash is $500
Second photo with 600D and 70-300f4-5.6L @$1350
As you can see it gets expensive with DSLRs and decent lenses and other equipment.
I'm not familiar with the capabilities of bridge cameras, but if you go on the likes of flickr there are groups dedicated to different cameras and equipment and you can see what's possible.
This was taken with the fuji finepix HS10 which is a fraction of the price
https://www.flickr.com/photos/melanistic141/7818985474/in/photolist-cUWkYo-qACT9E-aipmCZ-qx4xjh-rgASNe-fMGC8y-fmXuGo-nmJZFA-egMwhi-r7nPUk-cAErTs-d3mUZ7-gLQKM5-dfzPrp-fmnkEL-daLy2v-eedX2f-nnZ5XR-fHC8fA-cvW8cJ-nxjtbM-fv4EVi-ncV1nq-mrrU1t-frJgmx-dcAQQV-rG9WUz-drTpsC-foUreW-eUSzA7-chiDxw-czABA1-ezZdXg-en2y7V-cHVuFY-pFEoUt-nmfVYj-etMgyn-narNAa-cQyLLJ-mF8eYP-dgFoTk-mFak3w-nfWzGf-dh42bi-crJANQ-padTvi-cLZh1d-8vMmkB-fzq8DN