We started out with a £25 handheld fishfinder for depth (came with a transom mount and float for fishing) and have been up the Teign to the A380 a few times - although never launched at Town Quay. (We have thought about it

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+1 for VHF - best investment I made and both Cheryl & I have done the course - although she's a radio amateur anyway and used to do comms for Devon & Cornwall Police!
+1 too for NOT getting out on the Teign (or the Exe) - sand, shingle or rocky riverbed is fine, but both of these are mud and people have had to be rescued from both in recent years after getting stuck. You said yourself the tide goes quick - thus logically it comes in quick too - and if you were unlucky enough to get stuck in mud/weed, you could well drown before help arrives.
My best advice would be drop anchor, tilt up the motor and flush any mud you've picked up before it dries in the cooling channels and await the tide. We always take a few provisions, water etc so that's sound advice too - dehydration is a very real risk when out in a boat enjoying the sunshine and loosing track of time... If you have a decent length of chain on the anchor, you could perhaps throw into deeper water and with the aid of a boat hook pushing, you might be able to pull yourselves off. I have to say however that knowing the mud around there, I think the chance is slim. Don't all stand/sit at one end - you need the boat as level as possible to refloat at earliest opportunity. Don't risk using the engine either, you could very easily bust the drive or prop and end up in an even worse predicament

We've been delayed and arrived back at Polly Steps with barely a puddle remaining (old 14' ArrowFlyte) and as the bottom is stony there, we did get out and virtually carried the boat the last few feet to the slipway!

Cheryl had to get to work, otherwise we'd have definitely waited on the tide...
How do you get on launching at Town Quay? Looked quite 'harsh' in terms of rocks etc when we looked? (That said, we have launched at Steamer Quay in Totnes!)
Best regards,
Rob