Firstly, the NIP.
The prosecuting authority should send a NIP to the registered keeper (RK) so that, in normal circumstances, it arrives within 14 days of the alleged offence (the day following the alleged offence counting as day 1). They have obviously failed to do this. In itself this does not make further action invalid but it is open to challenge. The idea is that RKs should be told in reasonable time that their vehicle has allegedly been involved in an offence so that they can make enquiries whilst their recollection of events is still fresh.
Now, the six month limit. Speeding is a “summary” offence (meaning it can only be dealt with at the Magistrates’ Court) and there is an absolute limit of six months from the date of the alleged offence for the prosecution to “lay an information” before the courts. There are no exceptions to this. Since they are now offering you a fixed penalty and are giving you 28 days to respond you have until early June to accept or decline. If you respond after 17th May saying you decline the offer and want the matter heard in court I cannot see any way that it can be admitted for hearing as the six months will have expired.
Run this past your lawyer contact again to be sure of your facts and make sure you have not delayed the proceedings by (say) moving and not providing the DVLA with your new address. But it seems on the face of it they have made first class balls up and it seems to stem from the initial four months it took to serve you with a NIP.