You already have many good advice here and yes you need to match your "Expectations" with "Budget" knowing your sacrifices and then the gain you will achieve.
If you are looking for one lens with all capabilities at the budget or just around it, but with some given space for image quality loss and narrower F stop, your best choice is one of those ultra wide zoom lenses made by Sigma, Tameron or likewise. These are AF. Amazing optical range including good reach at upper end of 200 or 300mm but remember image tends to get soft at extreme end especially with wide open lens. Click on the links below to see more info.
Sigma 28-300
Tamron 28 - 300 mm
Sigma 28-200 Better diaphragm, lesser zoom.
Though you may get Tamron for cheaper, I suggest Sigma for better IQ (Image Quality).
Both company have different versions in this range with different prices so if you see a significant price difference check or manufacturing date etc.
You may go very cheap and go for same or close to this range optical but in Manual focus. You can save a lot of money but you need to be good to focus manually (Not a difficult task especially considering you are far from your subjects at the bank of Theme and then your subjects are slow moving ones.
But if you really need an extra ordinary reach, with excellent IQ and then most importantly a rather deceivingly small lens that people around you would not suspect your optical reach, I suggest this one:
LINK
You will love it except has no zoom. You can always sell it back on ebay for almost what you paid unless you go too bad in biding initially.
Or if higher zooming power and AF is a must check these:
LINK This one has a Vivitar Series -1 cousin that is the best choice for IQ, if the rang looks right for you. I didn't see on on UK market of ebay but you will see one if you look for it.
Tamron has a 200-400mm that also goes cheap, if you find a copy on ebay etc.
Then again another recommend zooms from me is an older version of sigma 70-200 that is not the newer EX version but the father of that lens. This has excellent IQ, on top of very wide diaphragm i.e. F/2.8 and I have a copy of this used on my D50 with no problem (Unlike the Canon version that has compatibility problem due to older chips). You need to search for this as it is not always available but I am sure you will find it if search enough.
All my suggestions are below, at or only marginally above your proposed budget.
Remember the relation between what you pay and what you get and hopefully you will do fine with your choice. Also remember as long as the lens work, world is not reached to its end and you can easily put it back on ebay. Even if you kill it, you can still get some money back on ebay or likewise by putting it up as "Untested", "As is", "For part only" etc. So go for what ever pleases you and be sure you are not going to get married to the lens at most it is going to be your girlfriend. If it doesn't fit your gadget or expectation, always there is an easy way to dump it.