Are prepared for the cost? It is very easy to spend £400 making a £100 bike. That may not matter to you.
This, many times over. I have just finished building a nice steel lightweight frame into a basic 5-speed road bike, and I dread to think how much it has cost me, compared to buying a complete s/h bike or even walking into a bike shop and buying a new one. And I was using the cheapest parts possible. To anyone other than me, it is worth buttons, of course. But I enjoyed the process and learned a lot on the way, so it's all good.
The other point is that it's actually harder than it looks. Like you, I have done a lot of maintenance and repair on bikes and thought I knew my way around, but I was always working on an existing setup that worked, usually as it came from the factory. BB length, rear wheel dish, rear axle length, derailleur capacity, chainline, all were worked out by someone else and built into a complete working bike. If you start with a frame, you have to work all this out for yourself, and it needs a fair bit of research. You could end up spending a lot of money in trial and error. I would recommend the Sheldon Brown website as a starting point, and there is a lot of expertise on this forum. My original plan was for a flip-flop fixie/singlespeed, but in the end I caved in and went for a simple geared setup. My frame has 130 mm OLD and it's near impossible to buy a wheel with a flip-flop hub in that size (I can't afford to have one custom built). I didn't want to cold set the frame (too nice and I wanted it to stay fairly original) and all the other options for getting a good chainline were bodges to some extent.
You can buy a flip-flop fixie/singlespeed on
eBay for under £200 brand new. It won't be great quality I would imagine, but if your urge for a fixie is just an itch, it might be worth trying that first. It will be cheaper than making your own, I guarantee it. I may well get myself on of these in the New Year as a present to myself, and get my fixie fix that way.
Also, as someone who grew up in Yorkshire, an Ellis-Briggs frame was the height of my adolescent fantasies. They are very, very good road and touring frames and rightly make a lot of money in good condition. The thought of anyone chopping one around to make something different makes me want to weep. You wouldn't buy a classic Jag with an idea of turning into an off-roader
Have a good think before committing yourself.