There is a lot of effort involved in making two passes in two directions so as to remove the vegetation that has built up on an old lawn and you may wish for the lawn care company to perform this these mowing tasks for you at additional expense.
The area will then be scarified hard and the debris cleared. It will be aerated with solid tines to a depth of at least 100mm and then again with over seeding tines and finally over seeded with new grass seed, fertilised with a slow release fertiliser and top dressed with a desirable top dressing material.
After about three or so weeks, the new seed will germinate and the new lawn will soon develop. It should then be fertilised again three months later and thereafter. The minimum clear run access required to move a professional mechanical equipment into a rear garden is 30cm so consideration should be given to chimney breasts, guttering down pipes, and out buildings that might obstruct access and brick walls and terraces allowing ease of access onto the lawn area.