For me, the signage is inadequate in that if one misses the sign due to a high-sided vehicle, there is no way of knowing the weight and width limit until one is committed to going onto the bridge. For me, a junction layout sign just before the traffic lights needs installing with the restrictions (weight and width) clearly signed for the ahead route.
Something similar to this sign showing the weight and width restrictions on the approach to the bridge on the embankment : -
https://maps.app.goo.gl/NfeCvLYmuVMrmGaGASo why nothing like this on the approach from Oakley St ? Easy one, I think. Embankment is responsibility of TfL, but Oakley St is responsibility of Kensington & Chelsea council who probably think, "well, every body knows Albert Bridge is restricted so why bother putting up expensive signs ?" However, they cannot escape their responsibility to erect 'adequate' signage, (Regulation 18 of The Local Authorities’ Traffic Orders (Procedure) (England and Wales) Regulations 1996 (LATOR)
Problem if you take them to London Tribunals is that your "inadequate" is the council's "adequate". It is a subjective term, so depends on how the adjudicator thinks when presented by the council with photos of the signs. A subjective assessment relies on personal opinions, interpretations, and qualitative judgments rather than relying on objective, measurable data. You would therefore need to show the sign that TfL have put up which is far better.