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"You will always be in our hearts and on our roads." - @Giro d'Italia on Michele Scarponi
@Shannonball
+Tom.Blackshire welcome to the neighbourhood!
As long as your bike has two brakes (and is generally roadworthy), then it is fit for purpose. Regent's Park is pretty flat, so you should be fine with one gear.
Swain's is doable on a SS/fixed-gear bike, but it is easier with a selection of gears.