This model just gets better and better - what an excellent signal box!
The discussion about levers, box design and Birkenhead Woodside reminded me that I have some information in various sources which, if not directly relevant to West Kirby Town, might be generally interesting. The sources are 'The Birkenhead Railway' by T B Maund (RCTS 2000), 'British Railway Layout Plans Vol 10 - LNWR in N Wales Chester and the Wirral' by John Swift (SRS 1996) and 'A pictorial record of LMS signals' by L G Warburton (Oxford 1972 but a reprint is now available). The Maund book is recommended for the sheer amount of detail and modelling opportunities.
Now for the detail...
The Birkenhead Railway was transferred to the LNWR and GWR on 1st Jan. 1860 and the Woodside terminus opened in 1878 when signalling was provided by the Gloucester Wagon Company. In January 1880, the block signalling regulations on the joint lines in the Birkenhead district were standardised as LNWR ones and from January 1885, the responsibility for signalling was taken on by the LNWR, so presumably any renewals or new work after then would be LNWR standard components. The main line was quadrupled around 1900 and a lot of new boxes were provided then. Although it was a joint line I think it always had an LNWR atmosphere.
John Swift's track plan of West Kirby shows a 63 lever LMS frame which includes the connections to the joint station so you may not need to worry at all about the exact number!
As for LNWR/MR rivalries in the new LMS, we might expect the MR to just assume that their arrangements would be the new standard, but just like the Horwich 'Crabs', Derby didn't get it all their own way. MR wooden signal boxes rotted at ground level, so the LNWR style brick bases were used. The MR hipped roof was too complicated so a simple gable was designed for the MR upper storey. Finally, the LNW staircase was thought to be too steep and so the MR gradient was used!