bowler always bowls from one side, there is no rotation
Note that this means that you are not playing in accordance with the Laws of Cricket. In particular, Law 17.1 states:
The ball shall be bowled from each end alternately in overs of 6 balls.
In theory, this means that the only answer to your question can only be "refer to the playing conditions for your competition", because the Laws cannot be used to answer a question when you are explicitly ignoring them. However, for the sake of practicality, I'll answer as for a match played in accordance with the Laws.
The important Law here is 18.12.1:
When a batsman is dismissed [run out], the not out batsman shall return to the wicket he/she has left but only if the batsmen had not already crossed at the instant of the incident causing the dismissal.
In the case you state where the non-striker is run out at the striker's end, the batsmen must have crossed1 so the striker does not return to their original wicket and they are now at what was the non-striker's wicket. In a regular match, this would then mean they face the first ball of the next over when bowled from the other end.
1. If they had not crossed, the striker would have been run out because it would have been their ground in which the wicket was put down; refer to Law 30.2 for details.