Here is a very quick look at the next 4 weeks using the JMA the Japanese Met Agency models. I have rotated the images and have added a red line so you can see the UK. Blues colours are low pressure, oranges are high pressures.
Week 1 19th to 25th
The UK will be mainly under low pressure systems, this will bring some unsettled and wet weather at times.
Week 2 26th May to 1st June.
High pressure to the North and East and South of the UK. This will bring some above average temperatures into the UK but it will be towards the end of the month of May before this starts to happen.
Week 3 and 4 3rd to 16th June.
This period looks like we will start to see some South westerly weather pattern starting to get in from the end of the first week of June.. Temperature will be around average but with above average rainfall.