It’s possible there are other questions like this around, but I wasn’t able to find an answer
Basically what are the actual halachos involved for someone who slept in and woke up after chatzos.
Do they say mah tovu, akeida, korbonos etc, hodi, boruch sheamar, ashrei etc, shiras hayam, birkas shma, and then when the amidah is reached, what then? Should one have in mind for minchah, then afterwards daven amidah again with intent to make up for shacharis? Should it be a nedavah, or a stipulation of nedavah? If so what about tachanun, is it said twice one after the other, or once after first shemonah esrei, and once after? Is ashrei said in between? What about a second set of korbonos?
You can say brachos, akeida, the korban tamid for shacharis can’t be said anymore, because it is after it’s time, however if it is already mincha time, then you can say it for the mincha’s korban tamid. You should say all of pesukei dzimra, because that is a praise to be said before the first shemona esrei of the day. After that, skip birkas kriyas shema, including kriyas shema until shemona esrei.
Regarding shemona esrei, the one you have to say at this point is the mincha one. You will have to wait until mincha gedola, which is approx. half an hour after chatzos in order to daven mincha. Regarding making up the shacharis shemona esrei depends on why you slept late. If you were an “ones”, or busy in a way that you couldn’t or forgot to daven, then you can still daven a make-up (called a tefilas tashlumin) shemona esrei for the shacharis one. However, if you didn’t daven because of negligence, such as going to sleep after it was already time to daven, without making sure that someone will wake you up, then you may not make up the shacharis one. If you are allowed to say a tashlumin, it is has to be said specifically after saying the current one (in this case mincha). You should not eat until you finished davening the tashlumin. You say tachnun between the shemona esei’s, then say ashrei, the tefilas tashmlumin, then continue saying lamnatzeiach, uva l’tzion, etc.
Regarding a tefillas nedava, it is very difficult for us nowadays to daven a tefiilas nedava, because it has a number of difficult conditions. The person has to know the meaning of all of the words being said, has to be able to concentrate for the WHOLE shemona esrei, and add a special request, in the tefillah. The poskim therefore say that nowadays we should not try saying a tefillas nedava.
 M:B 52-10
 Tzlusa Davrohom pg. 428, in the name of Maharil. As a side point, this would answer the question of Tehila l’dovid 108-1. Also see Eishei Yisroel 30 ftnt. 2.
 Since the zman of kriyas shem already passed. Regarding making up a missed kriyas shema, it is controversial if it can be done. See O:CH 58-7, M:B 58-29, nevertheless there is a preference to say it a second time after finishing maariv, see M:B ibid 28, Piskei Teshuvos 58-13.
 O:CH 108-1.
 Keren L’dovid O:CH -18, brought in Piskei Teshuvos 104-2 ftnt. 26.
 M:B 104-10
 M:B 104—12.
 M:B 104-13.
 Chayei Adam 27-17, Kitzur Shuchan Aruch 21-10, Pri Megadim brought in M:B 94-27.