Jousting armour typically had excellent neck and forearm protection. As you note, it would be very dangerous to not have it.

One example of one of Henry's jousting armours:

http://www.tudorplace.com.ar/images/HenryVIIIarmor02.jpg

I believe this is the armour he wore for jousting at the Field of the Cloth of Gold tournament.

Jousting armours could be field armours (i.e., armours intended for battle) with extra pieces of armour to provide more protection. On example of such additional protection is this grandguard:

https://collections.royalarmouries.org/object/rac-object-41076.html

which was made for this armour:

https://collections.royalarmouries.org/object/rac-object-11384.html

(among the 89 photos of this armour on that page, you will see some with this reinforcing piece, and other additional pieces, in place). The grandguard provides additional protection for the left shoulder and neck. The armour it is made for already has excellent neck protection. With this, it has even more excellent neck protection. The real-life Henry VIII would not have put up with the armour in the TV series!

Jousting helmets could also be made specially for jousting. For example,

https://collections.royalarmouries.org/object/rac-object-17505.html

and

https://collections.royalarmouries.org/object/rac-object-36557.html

(this type is often called a "frog-mouth helm").

Henry VIII took his combat sports very seriously, and could afford the best in available armour, and his armours show this. Two further examples of this are this armour:

https://collections.royalarmouries.org/object/rac-object-20.html

and his "spacesuit" armour:

https://collections.royalarmouries.org/object/rac-object-19.html

which is remarkable for its thorough coverage of the insides of joints (which are often unprotected, or protected by mail "voiders" in field armours).

Even with that superb protection, jousting was still dangerous. Henry VIII suffered a serious accident in a joust in 1536 when his horse fell on him, resulting in a leg injury and possibly a brain injury. The possible connection between the accident and his later tyranny has been discussed in various documentaries and web articles, but as discussed earlier, here, by u/rbaltimore in r/AskHistorians/comments/4i4e1w/was_henry_viiis_jousting_accident_in_1536_really/ and also elsewhere (e.g., Suzannah Lipscomb, 1536: The Year that Changed Henry VIII, Lion, 2009, who points out that Henry was on the way to tyranny already), it doesn't seem likely. An earlier accident, in 1524, where he was hit in his helmet while his visor was up, has also been blamed (see Lipscomb).