Greece's current bailout program runs out this month. Christine Lagarde told Greece that there wouldn't be a grace period. If €1.6 billion is not paid back by June 30, then it will be in default. The default is imminent. July will bring some interesting things.
Greece defaults on payment to IMF and ECB. The ECB pulls their emergency banking support. This leads to a run on Greek banks, capital controls and possibly even a Grexit. This is the most likely scenario.
The UK exports many products to Greece. Chief among these products being medicinal and pharmaceutical products (20%), manufactured articles and petroleum products. This may affect the UK quite a bit if Greece cannot purchase UK exports. Imports in Greece have slowed down. In April 2015, Greece's imports were down to €3.8 billion from €4.6 billion. A broke Greece means less income to the UK for exports. Greece also relies upon 17 million British tourists annually. The British government may be issuing travel alerts to British citizens in the event that Greece defaults. This will hurt Greece further if British tourists stop spending money there.
I can see the fire kindling already.