Living on a boat is generally seen as a very bougie choice and a lot of posh-ish hippie types do it, so people would probably think you had lots of money and fancied living on a boat. I think I'm the only person I've ever met who has lived on a boat and in council houses (but I'm sure there are lots of others, the stigma just isn't there).
A 15k budget for a long-term liveaboard boat isn't realistic though. Make sure you get a thorough boat survey and pay particular attention to any damp or rot before buying anything, as these are expensive to solve. You need to find a marina that lets you live there year round or get a residential mooring (a lot of marinas are seasonal and you can't live aboard over winter). This costs money. You need a boat licence for any waterways you want to use. You need a gas safety certificate to check your appliances regularly.
If you don't choose to live in a fixed place, or can't afford to, you have to move a certain distance over a period of time to be properly classed as "cruising" or you can be fined, and it's up to you to find a mooring at night when you're done cruising.
You will see cheap cabin cruisers (not houseboats) that aren't really suitable for 24/7 living all year around because they are not very well insulated or ventilated so due to the small amount of space in a smaller boat for less airflow, you will end up with a lot of condensation and damp (having lived on a 15ft cabin cruiser, everything was always damp). A lot of small day boats don't have toilets, so we were using a bucket in a cupboard. These days you can buy boat toilets quite cheaply though, so that's less of an issue (but factor the cost into your budget). Showers are harder to come by in small boats. You also need one that you can feasibly install a proper heater into, those little portable gas heaters emit CO which is dangerous for prolonged periods.
It's doable as a short term measure, but long term, you need to be thinking about saving money for your next move to a bigger boat (or somewhere else entirely).
For a barge, you will need to allow for cost and time out of the water (dry dock) for repairs and maintenance such as getting the blacking done.
You also need to sort out a PO box or someone to take your post for you, and you will need to sort out an address for your banking etc.
It's not as simple as just buying a little cheap day boat and moving on board, but if you do it right, it is cheaper all told than buying a house, but you usually need more money upfront and in this day and age, I don't think £15k will be enough, particularly if you're after a car with that money as well. £40k would be a lot more doable. BUT (and it's a big but) if you buy something small and cheap now, and look after it, then with the money you save on not paying rent/mortgage, you will have the cash to be able to upsize fairly quickly.