In the St Albans Diocese (Church of England) we have advised all churches to keep a written record of their FRA. This has been done on the basis of being the most efficient way of ensuring people think about what they do, to provide a record of what they have done and to assist future reassessments when required. Because the vast majority of churches have not carried out FRAs in the past, we have asked that every church (we have about 400 scattered over Herts and Beds) send copies of the FRAs to the Diocese so we can monitor how people are coping and identify any common problems. Churches, in their required 5-yearly inspections (the 'quinquennial survey'), will be asked if they have carried out their FRA.
As jokar says, the RR(FS)O applies to all places of worship and also to halls run by churches, mosques, societies and private individuals.
Re kernal's comments, the diocese may have more than 5 paid employees, and therefore Diocesan premises need a written FRA, but most local churches are run by a vicar or rector who has the 'freehold' of the 'living', as it is known. They count as self-employed for tax purposes and do not have a 'contract of employment' with the diocese, so I am told. But both HSE and insurers have recommended for some years that all volunteers at churches, whatever they do, are treated in the same way as 'paid staff'. Most churches have rather more than 5 volunteers by the time you add together sidesmen/women, cleaners, flower arrangers, choir members, churchwardens, bell ringers and tea-makers etc. so that's another good reason for putting it in writing.
With regard to being licensing for marriages I don't think that has any bearing on the matter - it's not to do with the 2003 Licensing Act regarding selling of alcohol and public entertainment which the RR(FS)O does mention. But of course any building licensed for marriages must be by definition a 'place of public assembly' which puts it fair and square under the new legislation.
One intriguing query for me is the status of 'stately homes' open to the public. If they are open to the public does that make them 'places of public assembly'? (I know the vast majority of these are well-protected anyway for the sake of the historic building.)