I live in a ground level flat with neighbours above. The basement under the flat is about 7ft high because we live on a hill. We use it as a basement for storage. We would like to convert it to an extra bedroom rather than move to a bigger house Q1. Who do we contact to find out whether the basement belongs to us. Q2. Who is our first point of contact to find out whether it would be possible to convert it into a bedroom - builders or planning department

Q1 Assuming you are a leaseholder then your lease should tell you this. We found that our basement was not in our lease and had to purchase the basement off the freeholder!
Q2 You'll need planning permission if you want to create a lightwell for example. The advice of a builder would be required to determine how far you could dig down (the depth are the foundations).
I would recommend speaking to somebody in your street who has down something similar. You should bear in mind that the cost of a basement conversion is high and may only pay for itself in areas where property is relatively expensive.

Deeds to ascertain the 'ownership' of the cellar. Talk to your upstairs neighbour about your plans, there may be a clause whereby you may not be permitted to alter the basement (foundations etc.)
Builders only follow Plans or Customer requirements so your first call is the Planning/Building dept.of your local authority for a discussion/feasability. Take photo's of what you are considering to have done to show the P.Dept.
Plans next if everything is given the Nod.
Builders quotes next, and it will not be cheap to convert your cellar to a bedroom.

Thanks for the advice. I now know my first port of call. No-one seems to have done this in my street so had been unable to get advice.

Note your comments about cost. I'm hoping it won't be more than £20k. Not too concerned about recovering costs in sale as it is more for practical reasons than investment. Glasgow prices are high so its too expensive to move.

The basement conversion on today's Room for Improvement cost £102K! And that was just for building work. There was then £7K for party wall work, plus the cost of fitting it out.