I don’t think you should just go ahead and carry out the work, as the common parts of the block are the landlord’s property, not yours. You could have problems if structural damage is caused, or someone has an accident.
Hopefully your leases each contain a covenant by the landlord to repair the common parts, in which case the proper course of action would be to sue the landlord for breach of covenant – however that probably won’t provide a very quick or cheap remedy.
Look into taking over the management yourselves by setting up a ‘Right to Manage’ (RTM) company – see
http://www.lease-advi.../faqs/faq.asp?item=65 for more information. The alternative would be to consider buying the freehold (leasehold enfranchisement) but that would probably take longer and cost more. Although I don’t have any direct experience I have spoken to a few leaseholders who have gone down the RTM route and it seems to solve a lot of the problems when landlords won’t carry out repairs, or charge extortionate amounts for sub-standard work.