You can expect almost everything regarding weather at the end of Agust except frost !!
I have read temp. and rainfall August 2005 at the Norwegian Meteorologic Institute (www.met.no) for Bergen:
average temp. +13.9 C
minimum temp ( night):+8.3 C
minimum day:14.0 C
maximum day : +22.1 C
Some rain ( from very little to rather much) in 20 days !!
I still wear my summer clothes in August with a light jacket/ windbreaker or rain jacket when necessary.
The Norway in a Nutshell Tour does not really need good weather - it is an all year tour. Of course if it will be foggy down the Flam valley - you will miss the view, otherwise it can be really nice in August. The boat (ferry trip) does not need good weather - the ferry will not go into open sea, so you will have no waves.
I think you will have a nice trip anyway.