With a POD no earlier than 1945 have the palestinian refugees overtrough the jordanian government and install the socialist republic of east palestine.
Going with the 1970 scenario suggested by Riggerrob,
Even if ethnic Palestinians rule the land, I doubt they would call it East Palestine, as they would be smart enough to predict that the Israelis would try to use a self-proclaimed Palestinian state as an excuse to deny any Palestinian claims west of the Jordan.
The ruling clique would probably call their state the Arab Democratic Republic of Jordan (or Transjordan).
Palestinian nationalism would not see Jordan as "home" but rather as a base for reclaiming Palestine west of the Jordan.
Would the rest of the refugees migrate or be deported to EP?
Again, in the 1970 scenario, no.
Palestinians uncomfortable with a left-wing state and who still possessed some personal property would not have an incentive to move voluntarily.
It would be state policy of the Palestinian-run regime east of the Jordan to discourage abandonment of communities west of the Jordan, although they would be glad to take in west bankers and Gazans and Israeli Arabs for military or political training to fight the Israelis guerrilla style. Likewise, at this time, the other Arab states' political commitment to oppose Israel is such that they would not want to signal at all that the need for a Palestinian homeland has been fulfilled. They would not want the UNRWA dissolved. They would deport only Palestinian individuals or small groups deemed politically dangerous, not set a blanket policy of mass deportation to Jordan.
As for deportation of Palestinians by the Israelis, it would take a bigger change than a Palestinian takeover of Jordan to make the Israelis willing to follow-through with a mass expulsion policy and the sustained, unflinching, mass violence that would be required to spur a 50-100% "successful" population transfer.
Likely the Israelis would only have the stomach to eventually go as far as they do in OTL, turning up the level of harassment and crowding of Palestinians, hoping that they move out (in the sense that lottery wins hope to win) but not really expecting it to happen.
Would this EP include Cisjordania and east Jerusalem?
Not in the 1970 scenario where the Israelis already occupy that.
How will this afect the middle east/islamic world/cold war?
Jordan is no longer friendly territory for the U.S., Jordan will by default draw closer to the Soviet Union, Syria and Iraq. But it will not exclude all dealings with western countries.