I was advised (about 2 years ago) that 6mm was ok up to 16m, and I'm
close to this, maybe a bit over.

Technically, 6mm can run this current, but only when clipped direct. Almost
any other installation method, the capacity is much less. The real problems
occur if you install within insulation. This is most likely if the cable is
buried in your loft insulation.

So it appears that the 6mm is not dangerous (assuming loft insulation
burial). However, it will get warm over a long period, as you have found, as
it is running so close to its maximum. It is temperature rise that limits
the cable current capacity, as the PVC insulation has to remain below 70C.

1) should I go to 10mm cable ?

Yes, if the run is easily accessible. You will waste a bit less electricity
and, more importantly, cable will run much cooler which is less scary and
will last longer.

2) I have purchased a 30mA RCD - but after reading about 'nuisance
trips' etc should I go for bigger ?

No. Use 30mA.

3) bathroom bonding is all ok, but one article recommends 'thickish'
cable for the secondary bonding line back to the board. Do I then
assume that the cable built-in earth wire ain't enough ?

The article is misinformed. You should not intentionally bond the secondary
equipotential bonding to the system earth. It may be implicitly bonded by
electrical or plumbing fixtures. This is usually unavoidable, but actually
decreases safety in some cases. (It does provide more reliable disconnection
of liveearth faults, but unfortunately provides a low earth loop
impedence, resulting in higher potential human leakage).

Christian.