Ring Doorbell - upgrade chime and transformer?
The amount of voltage and power that we are talking about here is too low to worry about stressing the wires or any other part of the system so don't worry about that.

The Ring will require a "modern" doorbell transformer, so that needs to be replaced for sure.  The old chime will likely not work with a 16v transformer if it was made for 10v.  It might work for a little bit, but it will sound funny and will probably fail quickly with the extra voltage.  What I'm getting at is you probably want to replace the transformer and the chime.  These are relatively cheap, so that shouldn't be a big deal.  

The existing wiring should be fine, but be very careful with it as to not damage the insulation.  That can get brittle over 60 years time so move the wires as little as possible.  The low voltage won't cause a fire or a shock, but it can ruin the transformer if the wires have cracked insulation and start to short out.