Considering the Junk Droid is not part of the group, I would go with "no" on this one.
Here is my understanding of the Junk Droid:
Ugnaught #1 activates, plays Spot Weld to put the Junk Droid into play in the Ready position. After the Ugnaught finishes his activation, the Junk Droid activates as part of his group, per Scrap Batallion. It is now exhausted.
Ugnaught #2 activates. The Junk Droid is part of his group, but can't go first because it is exhausted. The Ugnaught activates, and this readies the Junk Droid. Spot Weld would also move and ready the Junk Droid, but Ugnaught #2 does not need to do it unless he wants to warp it to him. After Ugnaught #2 does whatever he wants, the Junk Droid activates because it is Ready and activates as part of the group, per Scrap Batallion.
Scrap Batallion is a "game effect that allows that companion to activate as part of a different group" as mentioned in one of the bullet points in the Companion rules.
What this effectively means, is that you can have multiple Ugnaughts all remote controlling the Junk Droid across the map.
The Junk Droid is not a deployment card, so it does not ready during the Status Phase. It also does not have skirmish attachments, it is not part of any group (only activates as part of a group), and therefore does not share anything with any group.
If there were some way for the Junk Droid to be Ready before you activated an Ugnaught, you may even be able to activate the Junk Droid, then activate the Ugnaught and Spot Weld to move the Junk Droid and Ready it, then activate the Junk Droid again.
I don't think there's a way to get it Ready before you activate though, so this is probably irrelevant.
Though, I wonder what happens if you use the Ugnuaght and then it is killed mid-activation with Parting Blow. Maybe that's a way you could strand the Junk Droid in the Ready state? The group would leave play so the Junk Droid would stay in play but couldn't activate as part of that group anymore.
That seems to be what the rules say right now, unless there are some clarifications.
Written an article on my interpretation. would be grateful if you could check it out and see if you draw the same conclusions.
I have quoted rules taken directly from the rule books where necessary too.