Thames Young Mariners runs open water swimming sessions at their tidal lake which is more often used for youth water activities. Swim times vary due to the tide, because as certain tide times the lake is too shallow.
Sessions are £5 for 30 minutes but it is easy to email for an extended session. There are some safety rules about being observed on your first swim and what you wear but I have found them very reasonable.
Apparently there has been a seal in residence but I have not seen him, alas!
Red Bus Londinium Life 
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