First of all it's important to clarify which Paddington we are talking about: from subsequent replies I will assume Paddington (Suburban), but it could easily be Paddington (Circle). For ease, I will deal with the latter first.
This station does have passenger information screens (one on each platform) and they do work - fairly accurately - with the legacy signalling system. Whilst this area is a hole for realtime running information, it is covered by the 'Train Description' (TD) system which is set up from Edgware Road on the Inner Rail and Earl's Court on the Outer Rail. The TDs transmit generally without issue, and the running times are usually accurate - or as accurate as you can be with a 'distance' based system rather than real time running. Because the TD is an inherent part of the signalling system for automatic junction working at High Street Kensington (and to allow the Service Operator to know where to route the train at Edgware Road) it also means that the destination displayed is generally correct too (there's some exceptions, largely due to the fact that not every destination has an available TD!). Exact train location cannot be determined, but it is possible to get a vague idea using data from the Connect radio system (more on this below...).
Paddington (Suburban) and the rest of the Hammersmith branch is a little bit more complex. No TD system is in place west of Edgware Road (the Service Operator in the Cabin will set up and send them, but they never get received...), so trains are running 'blind'. As such, there's also nothing to drive passenger information screens from the signalling side of things, so it's difficult to know both where a train is headed (on the eastbound, at least!) and how far away it is.
At least, historically that has been the case. In preparation for 4LM (Four Lines Modernisation, the collective name for the upgrade of the sub-surface lines) new passenger information screens have been provided at all stations on the branch (including Paddington [Suburban]) and these have been slowly brought into use over the last year or so. The data is powered by a clever combination of the timetable and the Connect Radio system - every station has a radio cell, and every train is uniquely identified. The system knows roughly where the train is in accordance to the station, and combined with the timetable data it can marry the two together to provide a vaguely accurate countdown and destination. It isn't infallible and trains will often appear twice, or remain after departure, but it's an improvement over what existed before. If no data is being received then the screens will just display the direction of travel and 'Hammersmith & City line'. At Hammersmith there is a 'Next Train Indicator' which is manually controlled by the Service Operator from the Signal Cabin.
As mentioned, all of this will drastically change over the next month. It is anticipated that the first part of the new CBTC signalling will be switched in at close of traffic on 26th May, with Hammersmith (OZ) cabin closing and Automatic Train Control coming into use between Hammersmith and Latimer Road - SMA 0.5. With it will come realtime train information between those stations (inclusive), albeit limited as it will still be relying on legacy feeds east of Latimer Road. Eventually the information will be on a par (at minimum) with the other ATO lines (in particular the Jubilee and Northern lines). Resignalling should extend to Paddington (Suburban) (SMA 1) by the end of June - and by extension realtime information - with the Edgware Road and Baker Street areas (SMA 2) by the end of the year.
For the avoidance of doubt, installation of signalling equipment is broadly complete as far as Euston Square/Finchley Road/Paddington Circle (SMA 0.5-2), and operating passively to gather data. It is not providing any active control functions at this time, with all of the legacy systems remaining in place. Where improvements have been made these are separate, interim improvements and not part of the new control system.
And, finally: Edgware Road does not have a complex junction. One day the myth that Praed Street Junction is hard to operate will dissipate: it's very much the easiest part of Edgware Road (OP) signal cabin.