A large caveat with quantum resistant cryptography is that nobody can be certain that the new algorithms will never succumb to a clever new quantum algorithm. There remains the possibility of a new algorithm being created that has an impact similar to that of Shor's algorithm, but this time against one or more of the new techniques. Against this prospect, the new quantum-resistant algorithms draw on a greater number of mathematical problems for their security, partly to avoid putting "all eggs in one basket". We expect that this diversity will be reflected in the final NIST selections.
An interesting approach, for example adopted by Google in their early quantum resistant experiments, is to combine distinct pairs of algorithms to ensure that, even if one can be cracked, enough security is provided by the second algorithm (Google formed the CECPQ1 ciphersuite from combining a quantum resistant key exchange algorithm with a classical elliptic curve algorithm). We are likely to see this approach become more common, particularly if concerns remain about possible quantum attacks on the winning algorithms after the NIST effort concludes.
What advice would you give to organisations looking to prepare for attacks from Quantum Computers?
The main advice would be to start preparing now for the arrival of quantum computers: a lot of information which is secured today by vulnerable algorithms will still have value to an adversary able to decrypt it in 10 years' time using a quantum computer.
Be "crypto-agile": crypto agility is the ability to easily substitute one algorithm for another, without requiring expensive and time-consuming product changes. Preparations can be made by firstly surveying where and how vulnerable cryptography is being used in your information estate and to put in place the plans and capability to upgrade the vulnerable uses to quantum safe ones.
The hybrid, "strongest-of-two" approach mentioned above is an approach growing in adoption, combining the existing classical algorithms with a quantum-resistant one, providing the best of both worlds.