Why was the steam engine invented in the UK?

In the early 18th century Thomas Newcomen of Dartmouth developed the first steam engines to be put to industrial use.   They did the work previously performed by continuous shifts of men or horses labouring endlessly to keep floodwater out of Cornish tin mines.

A Newcomen engine at Farme Colliery, Glasgow in the early 20th century
(Glasgow Museums)

Newcomen engines were inefficient but effective, & expensive but long lived, remaining quite common until the early 20th century.   They were dismantled & re-erected as works closed & opened elsewhere.   There are several preserved around the UK, but I haven't seen one operating.

The basic principles :
Heat water in the boiler A, let relatively low pressure steam through valve I into the cylinder C.   Close I.   Some cold water from header tank L is then allowed into the cylinder C by opening & closing valve J, causing the steam in the cylinder to condense, so reducing in volume, & a partial vacuum is created in C.   Air at atmospheric pressure (about 14 lb/sq in) pushes the piston D down until the pressure in the bottom part of cylinder is the same (& the vacuum has gone).   The piston D pulls the chain E down, the beam rocks about the pivot F, pulling up the chain E, which hauls up the rod H.   H is attached to whatever is to be driven, eg lifting water up, or driving a wheel round.   Pipe M & valve N drain water so the piston rises for the next stroke.   G is an adjustable balance weight.   B is a safety valve.   Opening & closing valves could be done manually, but were usually done by subsiduary trip mechanisms (ie valve gear).

This early design of Newcomen engine was made very much more efficient & effective by James Watt, who added an extra condensing chamber - the driving cylinder then stayed hot instead of being put through repeated hot / cold cycles.

But why was the Newcomen engine invented in the UK?
Why was the steam engine developed & applied here? 
There were many difficulties faced.   For example, there was no coal in Cornwall, so it had to come from Wales, but it was easiest to sail round the Lizard and bring it up rivers such as the Dart.   Unfortunately, we were often at war with the French, who seized colliers & their cargo.   Some gentry did all they could to stop smelly & dirty coal, smoke & steam engines near their estates.   Meanwhile the government saw coal as eminently taxable & therefore controllable.   Commerce with its business relationships & power overcame the barriers.

For a fuller answer read Age of Invention: The Weight of Air.