Simon, only you can see what the circuit to the breaker panel includes besides what you've drawn (if anything).
Are there any additional connections to outlets at present, besides the lighting circuit on this branch? Or is what you've drawn everything on this branch?
Whatever the situation, to calculate this by watts (since that's how you described the light fixtures -- we usually use amperage in the States) check the breaker (or fuse if you guys still have them) for that branch, note the number of amps, multiply by your mains voltage to get totlal watts available before the circuit breaker trips.
Add up the wattage of everything you expect to use on that branch simultaneously, it should be a reasonable amount less than the tripping wattage. Otherwise you'll have the pain in the neck of resetting breakers. That lets you know if you have enough capacity. But doesn't necessarily address the overheating issue.
For that, you need to check your wire size for its rated amperage over the length of the circuit. As an example if a circuit is 100 feet long, and carries a maximum of 15 amps, I would know what size wire I need to carry that (here in the States -- usually by rule of thumb -- but also reflected in the codes). I'm sure there are British equivalents, but I don't know them. You need to find out if you want to do wiring yourself.
If a wire is too thin for the current it has to carry, it will heat up. It will also drop the voltage to a piece of equipment attached to it, and increase its working temperature. the longer the wire, the thicker it needs to be, also. Thicker wire IS more expensive, so generally you don't oversize wiring, but oversizing is a lot better than undersizing.
So knowing all of this, you do your homework, figure out what you have, decide how much more you can add, and what wire size you need, and wire it.
One other cause of electrical fires; poor contacts. Be very sure that when you connect wires to themselves or to terminals, outlets, fixtures, etc. that you use approved methods and tighten properly.
In a loose joint, over time, resistance builds up and creates heat, which makes more resistance, which makes more heat, etc. Then smoke. etc.
As Matt said, it looks like you are adding very little to your circuit with some flourescent fixtures, so very likely to be do-able. But be sure to do the above homework first, to make sure that it isn't already maxed out elsewhere, and be sure to use the right size wire, and make good approved connections.
Then light her up!