The registration of births began on 1st July 1837 in England and Wales, on 1st January 1855 in Scotland and in 1864 in Ireland. The records are held centrally in each of the these countries and indexes to these birth records are available online. The problem with using birth records for One-Place Studies is that they are indexed by registration district, which is a geographical unit that is much larger than a One-Place Study. It isn't possible to identify births that took place in your study area. Once residents of your place have been identified by other means, then births can be looked for on an individual basis. The information in the indexes is minimal and the cost of actual birth certificates, giving full details, is prohibitive for all but the most affluent OPSers. This is a case for co-operation with descendants of the residents of your place, who may have obtained copies of the certificates for family history purposes.
Civil registration of birth records in both the U.S. and Canada is a fairly modern record keeping system. Wander too far before the turn of the 20th century and you will not find civil registration of births. This is because the churches of the area were the primary record keepers. It was not until the desire to track disease and population did the governments intervene and request that the local civil governments take over recording these records. In the beginning, the record keeping was spotty at best, but after 1910, most if not all places kept civil registrations and sacramental records.
Check out the links provided to find more information.
http://www.freebmd.org.uk/Civil_Regis.html FreeBMD's information page about civil registration for the UK
http://bit.ly/1xFPXVd Page for information about birth records in Canada
http://www.archives.gov/research/vital-records/ National Archives general page for birth records in the US
Christine Sharbrough and Janet Few