Any local library probably has a collection of old Ordnance Survey maps of their own area, but there are a few libraries and other bodies that have maps going back centuries:
British Library, Map dept. This is the UK's biggest holding of maps, from the very earliest to modern day, and all available to consult (condition permitting). Everything is in their on-line search catalogue, but you can only look at 10 items per day.
Cambridge University Library, Map dept. This is also a legal deposit library, and has a vast stock of maps, from the very earliest to modern day, and all available to consult (condition permitting). Everything is very definitely not in their on-line search catalogue; you have to use their index cards in the Map Room to see what they really have. The maps, however, are all held on-site and are delivered very promptly. As it is part of CU, you have to get permission to access the Library (explained on their web site); looking at a rare map is probably sufficient justification to be allowed in. The Map Dept staff are always very helpful and knowledgeable.
Bodlean Library, Oxford University. This is also a legal deposit library. They now have all their maps in their electronic catalogue ("Solo"), but there will always be a long delay in providing the maps for inspection, as they are held off-site (in Swindon).
You can also scroll through their map card catalogue on screen, sequenced by Geography and sequenced by Mapmaker - there is even an autoscroll facility which is quite soothing. To find Essex you need to select the Alphabetic sequence, then E, then EN, then England & Wales Districts, A-M, then scroll (a long way down) to England & Wales Districts ESS - where Essex is spread over 3 choices.
Essex Record Office (ERO) This is Essex Council's archive of all matters for Essex, including a very good collection of antique maps. Their extensive collection is now available in their catalogue (called "Essex Archives Online" officially; it used to be called SEAX). The ERO is in Chelmsford, and is open to all (not just Essex residents).
The National Archive (TNA; what we used to call the Public Record Office). This has some maps in their massive holdings, all available via their on-line catalogue. TNA is in Kew, south west London, and is open to all.
There are several societies that specialise in various aspects of maps:
Charles Close Society (CCS). This society specialises in the history and collecting of Ordnance Survey maps. It has a three times a year newsletter, and has published an excellent series of books/booklets on each of the OS Series/Editions, plus other books on the OS. If you have a query regarding an OS map, this is the place to start. It's also very cheap to join.
International Map Collectors Society (IMCOS). This society is designed for lovers of antique maps, and works on an international basis. Its glossy journal is published four times a year.
International Society for the History of the Map (ISHMap). This is an academic society regarding the evolution of mapping - from earliest maps up to (say) 1800. Note that this website seems a bit erratic - sometimes it is not there at all and sometimes it is in Hungarian (it is located in Hungary), but in such cases try a day or two later and it is probably working.
Saffron Walden Historical Society. This is not specifically map-related, but it is a very new site just opened for the Society which has been going for many years concerning the history of Saffron Walden and its environs. It includes pdf's of many articles from the Society's Journals.
I have listed below sundry websites that I have found useful in some way in support of looking at maps.
Provincial Booksellers Fairs has listings of book fairs all around the country (about 50 a year); these generally have some maps on some of the stalls.
The London Mapfair is an independent fair organised by map dealers, taking place in June each year (at the Royal Geographic Society, near The Albert Hall), with a large number of dealers from outside the UK and high-end ones from the UK.
ETC Fairs has two types of monthly fairs in London.
See the Dealers page for a listing of map dealers.