While you could use BioMart or the API to retrieve these data, you might be better off with VCF file dumps:
Homo_sapiens_somatic.vcf.gz (all somatic mutations from COSMIC and dbSNP) and Homo_sapiens.vcf.gz (all non-somatic variants from various sources, see http://www.ensembl.org/info/genome/variation/sources_documentation.html ) from this directory should be a good place to start.
If you specifically only want those from the 1000 genomes project, you could grep for the 1000 genomes evidence flag:
curl ftp://ftp.ensembl.org/pub/release-79/variation/vcf/homo_sapiens/Homo_sapiens.vcf.gz | gzip -dc | grep E_1000G | head
or download the population-specific files from the above directory, or the VCF files from the original project site (ftp://ftp.1000genomes.ebi.ac.uk/vol1/ftp/release/20130502/ ).