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I have a smallish patch of lawn - approx 30m x 8m which is reasonably
level, but has the odd bump and dip. With children in the house it
gets a fair amount of abuse, and has hardy (non-rye) grass. However
due to neglect on my behalf the quality of the grass is deteriorating,
with weeds appearing, and grass thinning in places.

So in proper money, approx. 100Ft by 25Ft.

For this size you will want a petrol mower. Smallest and cheapest size are
16" push machines. Within your budget, you can forget about choice and
options, it will be a 16" push machine four wheeled, non mulching, rotary
mower. Which TBH for want you want to achieve is probably the right machine
anyway.

Now two statements in your post,

"This isn't helped
by the rubbish "own brand" rotary mower bought from a shed"

"I don't want to buy
from a shed again, but the last time I spoke to an independent mower
specialist he tried to sell me a £200+ petrol jobby"

One question to ask first, when you go and buy a car, television, washing
machine or vacuum cleaner, do you go for the cheapest possible one
available? Nope, probably not. When it comes to garden machinery, this
mentality seems to kick in.

There is a reason that the mower you bough from a shed was rubbish, because
it was cheap. There is also a reason that the indepedent tried to sell you a
more expensive machine, because it isn't rubbish and better quality costs.
There are other factors, but I'm sure you get the point I'm trying to make.

You can go to the sheds and get a push petrol mower for £99, probably made
in Europe but with a Chinese engine on it. The deck will have not material
in it (i.e. it will be the thickness of a silver foil wrapper) and the
engine will be the quality of toffee on a hot summers day. Forget about
spares, the machine is replaced in warranty, after that it's scrap.

You could go to an independent, who will have a 16" push machine for
£150-£170, which has a decent deck and probably a Briggs & Stratton engine.
It will be better built, last longer, start easier and have spares
available. Plus if you have a problem, you'll have someone to go back to.

You pays your money and takes your choice.