Silver and brass have very similar melting points, around 950. Nickel Silver doesn't actually contain any silver, it is mainly Nickel, much higher melting point at 1400+. I regularly work with all three and for joining parts regular electrical solder works perfectly. Plumbers flux also works well, in tiny amounts and with the very important caveat that you need to be able to wash off the flux residue immediately afterwards.
A gas torch is generally much better than an iron. I use one of the little ones intended for cooking. they are less than £10 and run on gas lighter fuel. Adjustable flame so you can get it very fine.
The trick is to get the relevant area up to temperature as quickly as possible, with minimal spread of the heat outside the area actually involved. So get your parts together and fluxed and place small slivers of solder on the area you want to join. Then apply the flame until the solder melts and runs into the joint. Don't be afraid of the heat. Generally a very hot flame applied for a few seconds is better than a low one for much longer. Bear in mind the solder will go wherever the flux does, so apply that sparingly. The plumbers flux is water soluble, so you can make it slightly thinner and apply with a small brush, or the tip of a needle for very small parts.
Generally you want to have the solder in place before you heat the parts up. Applying it from the end of the coil is fine for plumbing or electrical work, but results in a lot of work to remove the excess of you are doing jewellery or watch/clockmaking, where appearance is important and you don't generally want to see the solder.
Worth having a selection of solder. For soldering round parts it can be very good to use very fine solder wire to make a hoop that will drop neatly on where it is needed. For really tiny stuff you will find if you cut a tiny sliver of the end of a flux cored solder, you end up with effectively a tiny washer of solder with a hole in the middle, can be very useful.
