Grass will grow on most soil types in the UK. It is more important that the soil is not compacted. This is because the roots will not penetrate the soil nor will there be any tiny spaces in the soil for nutrients, oxygen and water to accumulate.
To de-compact and aerate soil you need to spike it with a hollow tine fork or if you do not have one of these an ordinary garden fork is a good substitute.
Soil Ph does affect grass growth. The range most lawn grasses will tolerate is between 4.5 and 7.5 on the ph scale. If you have acid soil fescue and bent mixtures, e.g. Luxury Lawn and Shady Lawn, will tolerate these best. Perennial ryegrass mixtures e.g. Quick Lawn or General Purpose are not so tolerant to acid soils usually requiring a ph above 5.5 or 6. It is best to experiment a little if you can because despite all the data grasses can persist in adverse conditions.
Sandy soils tend to drain better but lose nutrients easily so lawns on this type of soil may need more frequent doses of fertiliser and water.
Conversely heavy soils can be waterlogged in the winter and rock hard in the summer. Adding sand or loam to these soils on a regular basis can eventually improve them, however, this brings us back to the beginning to the benefits of de-compaction and aeration.