Budget TT bike

Shouldn't be a shortage of TT bikes as they didn't sell like all the other types during 2020. In fact they were the only bikes being sold with discounts at the end of the year. The issue with the likes of Ribble are the component shortage. You will struggle to find anything at 2k though. TT bikes are always a good second hand option as they've often had very little use other than some turbo and a few races. A classic would be a 2015 onwards cervelo P2 or P3, which you could then trick out with aero gadgetry as you get more serious.

As said above, maybe a road bike could work... for example, my Argon18 Nitrogen has an aero frame and also a reversible seat post top to get you in a forward TT position. Fit some deeper section wheels and tri bars and away you go!  I've seen Nitrogen's on eBay to allow scope for you to get in close to budget, as well as having a versatile bike for normal use. 

Unfortunately you're probably shopping at the wrong time - between the shortage of new bikes due to Covid and the arrival of Spring and resumption of racing they will be harder to find. Secondhand should normally give you more options.

How serious are you? Would you consider a road bike build with base bar, aero extensions and bar-end shifters? If the frame size is suitable then it can get you surprisingly close to the aero-ness of a full bling TT rig.

I'd argue that fancy stuff like hidden cables, latest generation kammtail frames and integrated front ends are not the main things that help you go faster. Getting yourself into an effective aero position is far more important.