I agree in general with the above, but having done the engine, box and diff swap from 803 to 948 myself, I'd note the following.
1. It's often said that the 803 box is weaker than the 948 box - having had both in bits to convert a 948 box to take the 803 gear lever, I couldn't see any difference in terms of the strength of the innards. the main difference is the ratios of 1st, 2nd and 3rd gear (which are terribly low and close on the 803 box) - 4th remains the same (at 1:1) so doesn't impact on revs in top gear. The 1098 box is definitely stronger, with baulk ring sychros and ribbing to give the case additional strength.
2. I agree that it is, however, best to change the box to a 948 one, as the ratios are far better suited to the engine, with nicer spacing between the gears. As mentioned before though, it is wise to change to a 4.55:1 (948) diff to get a vast improvement in 'cruisability'. Personally, I wouldn't go for a 4:22:1 diff from a 1098 with a 948 engine,as it isn't torquey enough - ampwhu has said he's got a 4.22:1 in his car with a 948 engine, but I think that might be an A30, which is quite a bit lighter (sorry if I've got that wrong, ampwhu).
3. Regarding the question about the reverse selector - in the 803 box, you have to lift the lever to engage reverse, but in the 948 box, there is a spring to prevent inadvertent selection of reverse in the remote tailshaft - once you replace this with the 803 tailshaft, there is nothing to prevent the gear lever from moving across into the reverse gate. Hence it's best to remove the reverse selector form the 803 box and replace the reverse selector in the 948 box with it - this way, you preserve the requirement to lift the lever to engage reverse gear.
4. The 1098 box can be adapted to take the 803 tailpiece and gear lever, but it requires machining work. A few people on here have done it, but it is quite involved. Have a search for it - I think it might have been 'win' who did it but I could be wrong.
5. It's a good point regarding brakes - as 7" front drums were also standard on 948 cars, you shouldn't have any trouble with insurance if you change all the running gear to 948, as the brakes of a Series II are the same.
6. I did mention it before, but in case it's been forgotten - if you do change the box, you will need to remove the speedo drive from the 3rd motion shaft of the 903 box and fit it to the 948 box. That way, you can screw in the correct speedo drive to the new box (i.e. the external piece to which the cable attaches) and your speedo will read correctly. If you don't do this, it will read around 50% too fast

. To be honest, that is the only part of the process that is a bit of a faff.
EDIT: Palacebear - just going through the previous posts, as I guess you've already sussed out, the easier route for you regarding the speedo would be to use one from an early Minor 1000 - that way, you don't have to mess around stripping the gearbox to replace the speedo drive and you can still have a gold-faced speedo which will read correctly. Fitting a 948 diff will in all likelihood make the speedo more accurate, as they mostly over-read wildly as standard. When I did this, it made it fairly accurate when checked against satnav (as was also the case when I fitted a 3.9:1 to my 1098 car!). Finally, I don't think anyone answered the question regarding the carb - the 948 has an 1 1/4", whereas the 803 carb is 1 1/8". You'd be well advised to swap it, but I am certain you'll be able to fit the SII air filter which will keep it looking original.