The UK (along with Ireland) does not participate in the Visa Information System:
"As a Schengen instrument, this regulation applies to EU countries with the exception of the United Kingdom and Ireland. Denmark has decided to implement the regulation which also applies to Iceland, Norway and Switzerland."
Thus it should not have any access to data held in the VIS.
However, should you inform the UK authorities you have been refused a visa in France, no doubt they will ask the French authorities for any information they hold on you.
N.B. There is also the Schengen Information System - a different database - to which the UK does have access. However, unless you have been banned from entering the Schengen area, your data should not be stored within it (as far as I understand).