I've been asked by the local authority who licence the property to supply an 'Electrical Inspection Condition Report' for my 1 bed flat.
Is this the visual inspection or one that tests the electrics?
Does anyone know approx. costs for these in East Sussex?

Bit late in the reply and I assume you'll already have dealt with this. But for anyone else, this is a condition report - not just a visual inspection.
The electrician will inspect various aspects of the installation - he'll look behind a sample of the sockets and light switches, how many depends on what he finds and his "gut instinct" as to what sort of problems might be lurking. He'll also do some tests using specialised equipment. At the end, he'll do a report that will list a load of numbers, but the key bit you will be interested in is one box where it says whether the installation is "satisfactory" or "not satisfactory" - the latter being fairly self explanatory !
There is also a section to list any defects/problems which are coded for severity.
C1 is the "this is dangerous and needs immediate attention" category - get them fixed right away. This might be things like broken/missing covers exposing live parts, or faulty/incorrect components, and so on.
C2 is "it really wants looking at, but isn't IMMEDIATELY dangerous". Again, these need fixing, but not necessarily NOW.
C3 are mostly "not to current standards" - ie they were OK for the standards applicable at the time, but aren't now. Two really common ones these days are that the consumer unit is plastic (regs required "non-combustible, eg metal - but without actually specifying what level of non-combustible they meant - since January this year), or cables buried in the plaster that aren't RCD (or otherwise) protected (a requirement that came in a few years ago). For these, dealing with them is optional since the regs aren't retrospective.
However ... I work on the principle that if there is something on the EICR then that's something that a devious tenant might use to claim that "the electrics aren't safe", though I might well leave the plastic CU since I replaced them all not that long ago (to upgrade to all RCD protected). If you have an EICR with now defects listed and condition as satisfactory then it would be hard for anyone to argue that the electrics were faulty.
As to cost, it varies considerably. I recently had one done on my own house as it was being sold, it cost me £150 (inc VAT). Realistically I wouldn't expect a genuine one to cost much less due to the time required to do it properly. If you have a big property, or a complicated setup, then it may well cost considerably more.

Now, there is one issue that's difficult to quantify ... On another forum, there are occasionally questions along the lines of "are these genuine faults ?" Sometimes it is rather obvious that the electrician doing the EICR is either being unduly picky, or plain dishonest, in order to drum up some ore work - often quoted at "above market rates". For someone unfamiliar with the regs and terminology, it can be easy to be misled by someone who appears to know what they are talking about. Certainly if someone tells you that you *MUST* fit a new consumer unit because your current one is plastic then you know they are lying - but you need to be careful not to misinterpret as they might be strongly recommending if yours isn't fully RCD protected and being plastic is just one more reason.
If in doubt, head over to diynot.com and ask in the UK Electrics forum there. Include as much information as you can, and ideally photos of the items mentioned - eg if you are told that you must replace your CU, then include a photo of it so that those who do understand these things can see exactly what you currently have.