The pros of a managed third stage are quicker third stage, less chance of PPH. It helps the uterus clamp down to expel the placenta which can mean less chance of a retained placenta (the uterus cannot clamp down and seal off the blood vessels properly with the placenta still in there) Baby can be cleaned and weighed more quickly as the cord is cut and baby is therefore more mobile!
Downsides - less time for placenta to come out before the cervix closes, baby doesn't get the full amount of cord blood and this can mean lower iron levels, one study has shown that prophylactic use of syntoncinon (one of the ingredients in syntometrine) can reduce it's efficiency next time it is used - ie if you have it in your first birth, then it is slightly less efficient (so it's not necessarily a good plan to give it to all women 'just in case' but that needs further research)
The pros of a physiological third stage - fewer drugs into your system, baby can be left attached to you for as long as necessary, breastfeeding immediately should help with the detachment of the placenta, you can keep baby next to you without anyone suggesting they take baby now to weigh/clean/measure etc. The placenta may take longer to deliver but you reduce the risk of the cervix clamping down with the placenta in place leading to retained placenta. Retained placenta is a cause of PPH.
Downsides - Chance of PPH due to the uterus not expelling the placenta quickly. Baby is attached to you until either the placenta is delivered, or the cord is cut, so you can't pass him/her to anyone else or move without baby being there.
Prof. Cecily Begley at Trinity College in Dublin has done research into managed vs physiological third stages, and has found that there is no increased risk from waiting and seeing if the injection is needed to reduce bleeding instead of giving it at the moment of birth. Even if it's given later on if the placenta hasn't delivered, it will still only take seconds to do it's thing.
Most MWs will want the placenta delivered within an hour of the baby being born, however the AIMS book 'Birthing your Placenta' states that this one hour limit is seemingly arbitrary and not actually based on any evidence - Mary Cronk has said that she's been with women who have taken up to 3hours + (IIRC) to deliver a placenta naturally but have done so safely.
I had a retained placenta which was manually removed AND a PPH first time round, and am opting for a physiological third stage this time unless I have another PPH. I did have lots of intervention in my first birth though which could have contributed to these things happening, and so I have planned a homebirth this time round to try to minimise intervention and therefore reduce the risk of PPH and retained placenta further.
So in short, yes, you can wait for the placenta and then have the syntometrine jab after a while if you want it.