Is there a restriction on the gap you can have between spindles?

Yes. The maximum gap between spindles, and in fact any part of the staircase must not exceed100mm. This applies anywhere where children under 5 years old will be, to ensure they cannot fit through the gap. For new homes and extensions, It is also a requirement of Building Regulations Part K - A copy of this is obtainable from the following link https://webarchive.nationalarchives.gov.uk/ukgwa/20141202115025mp_/http://www.planningportal.gov.uk/uploads/br/BR_PDF_AD_K_2013.pdf