The eagerly-awaited study, was originally due out in the spring but will be held back until after the local, mayoral and European elections, according to the HLF.
The Government issues guidance to civil servants and local authorities not to release anything that could be politically sensitive in the run-up to elections - a time known as "purdah".
While it makes decisions independently, the HLF is a non-departmental public body that reports to Parliament through the Department for Culture Media & Sport.
The elections on 22 May will combine voting for European Parliament representatives, more than 90 district councils in England, all metropolitan district councils, all London boroughs and some parishes. Five cities will elect new mayors.