Do you think it is worth putting a few inches of top soil over the who
garden? Would this be expensive?

That's effectively what will happen if you top dress regularly, as Pied
Piper suggested. You don't do it all in one go - you add a little in stages
each autumn. A normal lawn top dressing is a fairly sandy mixture of
consisting of sand, sieved soil and peat/peat substitute or leaf mould. Over
time this will improve the quality of your lawn particularly, I think, if
you were to hollow tine before the top dressing which would remove plugs of
the heavy clay.

If you are looking for a quicker solution, I guess you could start from
scratch. Rotavate the whole area. Double dig to remove any "soil pans" which
could be part of your drainage problem and incorporate lots of manure to
improve the soil structure. Then returf or reseed. You could have a nice
lawn by the autumn, but it would be hard work and relatively expensive.