about Cloning
the Cell nucleus cloning happen first in 1996 with Sheep Dolly
before that there was embryo cloning, here a very early embryo is disjointed in single cells, each start to growing them self into a Embryo.
This technic was discover already in 1970s in combination with In vitro fertilisation and you can have clones in 1977
The Nuclear War
what you need is a atomic conflict who not destroy Civilization, so a Global nuclear is no go for this TL
like Mr_ Bondoc point out the Sino-Soviet border conflict erupted in war is good base for limited nuclear conflict
that include with nuclear winter scenario and we got dark and dirty sky from movie version.
On Space Flight
with Earth ravage from effect of nuclear conflict, Mars become very interesting real-estate
you don't need Dyna-Soar orbital bomber
what you need is very heavy lift rocket to get Colony equipment and colonist to Mars
like Boeing MLLV with payload of 3.5 million lb. or 1587 tons in low orbit.
because the world is already radioaktive contaminated
they could use ORION Nuclear Puls engine with ground launch that bring 8.8 million lb. or 4000 tons in earth orbit.