I thought I would put a few places down that I have stayed overnight and launched either a boat or kayak, many have slipways and all have shore access.

1.Sailing/boat clubs that you may need to join as a temporary member for a nominal fee.

Netley sailing club, Southampton.
Dorchester sailing club, Oxfordshire.
Weymouth and Portland sailing academy, Dorset.
Hythe Marina, Hampshire.
Mylor yacht harbour, Cornwall.
Loughour Boating club, Swansea (on the CM listings)
Towy boat club, Carmarthenshire.
Clywedog Sailing Club, Llanidloes. Mid Wales.

2. Campsites with launching facilities or next to beach.

Pentewan sands, Cornwall.
Redcliffe farm, Dorset.
Fishery Creek, Haying island.
Newgale sands, Pembrokeshire.
Shell Island, Gwynedd, North Wales.

3. Car parks where overnight is permitted.

West beach, Hayling island. (on CM listings)
Exmouth Promenade, Devon. (on CM listings)
Churchfield carpark, Appledore, Devon. (on CM listings)
Riverbank carpark, Bideford, Devon. (on CM listings)
Burry Port. Carmarthenshire. (on CM listings)

These are just places from my memory, there may be more that come to mind as I think about it.