Hello and thank you for your question. I will be very pleased to assist you. I'm a practicing lawyer in England with over 10 years experience.
From what I understand you want to install a kitchen and in order to make use of the existing drainage pipes you would need to install pumps and would therefore prefer to lay a new pipe across a neighbouring lawn. Could you kindly confirm I have understood this correctly please?
The first step is to consult your title entries. Assuming your property is registered at H M Land Registry you can obtain these online if necessary. http://www.landregistry.gov.uk/public/property-ownership
Your goal is to check the title to see if you have any pertinent rights to drain over the neighbours land. Such rights if they exist will be evident in the A register (Property Register) on the title. Normally you would expect to see rights of drainage to deal with existing pipes. It is less usual to see rights to lay pipes but they can exist.
What you are looking for is the right to lay conduits and pipes over the neighbours land. As above do not expect to see this as it is not usual for such rights to be included unless land is being sold for development but if such a right is there then it can be enforced.
If there is no such right which is unfortunately the most likely scenario then you are faced with negotiating a new drainage easement with the neighbours. You would need to have agreement from any neighbour that has rights to enjoy the garden and also the freeholder of the block of flats. If such persons will agree then a deed of easement can be prepared and signed and then registered on your and their titles. This will give you a right to lay the pipe and drain under their garden.
This is largely only as complicated and expensive as the parties involved choose to make it. Potentially all parties could insist on independent legal advice as well as a premium which could potentially make the exercise very costly. Alternatively they may be very willing to agree and sign without a premium (or minimal premium) and without legal advice in which case you would only need to bear in mind the cost of your own solicitor and registration fees at the Land Registry which would likely be a few hundred pounds in total.
There are no laws which enable you to enforce a right to drain over their land. There are statutory rights which enable water companies to lay pipes without landowners consent where necessary subject to payment for damage caused and so you may be able to have a pipe laid by a water company if they will not agree but this can only be done where it is necessary and if there is an alternative drainage system already in place the water company may not be able to help. You can ascertain by contacting them if you have not already done so.
Does the above answer all your questions or is there anything I can clarify or help with any further?