Yes, the voluntary transition away from lead shot for live quarry shooting was voluntary and many shoots have made the move including the GWCT on its Loddington shoot.
A quote from the GWCT response: GWCT Director of Policy Dr Alastair Leake added: “Scientific monitoring of wildlife over the past 30 years at the Trust’s demonstration shoot shows game management delivering a clear biodiversity net gain and playing a vital role in reversing the decline of many red-listed species. The shoot has easily adapted to non-lead ammunition over the past two seasons, which will enhance its positive impact.”
A further insight from GWCT's Mike Swan on Loddington and other shoots moving away from lead shot here:
www.gwct.org.uk
Here is a comment piece by CA following an article in Shooting Times about lead shot ingestion by grey partridge adults and chicks - the latter being a key concern as regards chick survival rate
Countryside Alliance analysis of research about grey partridges dying of lead poisoning.
www.countryside-alliance.org
The Shooting Times article is here:
Wild grey partridge chicks face many challenges from the moment they hatch. They start out the size of bumblebees and, with lemming-like tendencies, the parents have their hands full getting their offspring through the critical first few weeks. We all know the story of the English partridge and...
www.shootinguk.co.uk
As for the data, the 2005 GWCT paper on grey partridge is here:
A paper on population modelling based on available data in UK and France is below. The data itself showed lead shot ingestion in grey partridge as 4% of direct causes of mortality and 7% of ultimate causes of mortality. As for the modelling this showed that lead shot ingestion at modelled rates reduced population size of partridges by 10%, and when combined with bait and pesticide poisons, by 18%.
Little is known about the magnitude of the effects of lead shot ingestion alone or combined with poisons (e.g., in bait or seeds/granules containing pesticides) on population size, growth, and extinction of non-waterbird avian species that ingest these substances. We used population models to...
pubmed.ncbi.nlm.nih.gov
It is now widely accepted that the main cause of the grey partridge decline in the UK was due to reduced availability and density of the right insects at the right time for chicks. Outside of partridge manors, this was not well understood until it began to be studied. Acceptance of the impact of herbicides and insecticides came later.
Lead shot ingestion and its impact continues to be studied. Understanding and acceptance will come later as per attitudes to herbicides and insecticides. I don't think lead shot is the cause of the grey partridge decline, but it is likely to be amongst many factors hindering its recovery, and moving away from lead shot will help grey partridge and many other bird species that eat lead shot as grit or mistake it for seeds, as per the GWCT response in the OP.