If you have a promising young leg spinner in your team is it better for their confidence to bring them on to break a partnership or straight after you get a wicket when a new batter comes in?

Depending on conditions, I believe a captain should always back a promising young leg spinner and not hold them back for any particular reason. Support them with the appropriate field settings and encourage them the flight and rip the ball. Be positive in attitude towards the young player and instill confidence in them. They have been selected for a particular reason in the team. If the captain is not confident in the player then selections must be questioned. Young spinners The best way for a young player to learn is by bowling as much as possible in game situations.

A very good question. A supportive captain is so critical in the development of young spinners and it’s so important the captain backs their spinners at all stages of the game.
In the scenario you’ve mentioned there’s a couple of things to think about. How many games the leg spinner has played in the team, their level of maturity and personal confidence.
Ideally, it’d be great if the young leg spinner came into the attack straight after a wicket falls so they get to start their spell and build their confidence against a new batter who may take some time getting used to the bowling, the pitch and the conditions.
However, in my opinion the quicker you get the spinner into the game the better.
Ultimately the captain needs to do what’s right for the team and if the partnership continues you may not have a choice but to bring the leg spinner on.
Get them into the game, provide some protection in the field and if the batters go after the leg spinner you can always give them a spell after 2 or 3 overs. If you do, please make sure you communicate to the spinner and ideally look to get them back on again as soon as a wicket falls.
Leg spinners are match winners.

Agreed, I found a good way to allow a spinner to relax and get into the game was to tell them they have X amount of runs to play with. Regardless of what their first couple of overs went for they didn’t fear being taken off.

I find it depends on the situation of the game and the batters who are in. if the batters are scoring at a 200 strike rate, probably not the best time as they will probably 'smell blood in the water' if a young leggie is introduced.
if there are defensive or new batters in, that's probably a good time so they can get their rhythm and confidence.
if the game is meandering along, it might be a good time to introduce them to break a partnership or the batters rhythm to create pressure for the other bowler.
Also, if the batter uses his feet and hits them for a boundary off a good length ball, this is also an opportunity to compliment them and tell them that is what you want to see the batter doing and encourage them to keep bowling there or perhaps vary their line or length slightly so the batter doesn't get quite to the pitch of the ball and mishits the next one.
lots of variables but young leggies in particular need to know that you expect them to go for a few runs, it's ok, and to keep trying to hit good lengths and take wickets.
the last thing you want is them to do is lose their confidence and not know where they are landing the ball.
give them protection on the fence if they need it but always let them feel you have confidence in them.