Yes, Hospital for Tropical diseases in London/Liverpool is supposed to be the best place in the UK NHS for these things, but you need a referral to see them from another doctor first. They say they have a "walk in clinic" in London but I tried and they refused to see me and said they only would see people in the walk-in clinic who had travelled within the last 6 months and have symptoms of high fever etc. such as that caused by malaria.
I think it would also be interesting to hear of any other private labs that people in the UK have used, as I had real problems funding such labs? Anyone used anyone other than that one in Leeds that you mentioned (though it doesn't seem to cover a very wide range of parasites, only the protozoans)? I also had tried to contact a lab called "The parasite Clinic" in London but they never answered an email I sent them so not sure if legit or not.