I am thinking of changing from the standard Commer deep dished low compression pistons to standard dished ones (18861 ref) to increase the compression and raise the power output.

This looks like it should change the compression ration from about 7.5:1 to 8.4:1 or so.

I have a MM stainless exhaust and 34 ICT Weber carb fitted. The rest is all standard.

Will this help get more power, or will I have to add a better carb set or change the cam also? Any advice from those that have tried this appreciated.
All those things will help but I think an SU conversion with a generous head skim would be the easiest. Tim (commeracer) recommended around a sixty thou head skim for best results.
I've got a Rapier engine in Harvey the Auto-Sleeper with the SU and he goes very well (with a few other mods)
I did an SU conversion and one of Martins exhausts on Bonnie the Jennings (everything else on the engine is standard) and the transformation was well worth the effort.
I spoke to my local engine reconditioning specialist in Cardiff who said 60 thou was a lot but importantly not reversible. He recommended changing the pistons to gain compression in case the mods don’t work out, which seemed sensible.

The SU conversion looks a very good modification too. But I have always wondered how you get oil in the dash pot with the carb sitting underneath the passenger seat. I seem to recall having to fill mine up weekly when I had SUs.

I guess first stage is to get the head off and see what condition everything is in. The Commer had an engine rebuild in 2004 (including a new cylinder head) so I suspect not too bad as she has only done 5000 miles since then.
Yep 60 thou is a big skim but it seems to work and it's unlikely you'll ever need to skim it again, send Tim a PM and ask for his views - what he doesn't know isn't worth talking about. The SU sits neatly under the side engine cover which needs to be removed to top the dash pot up - but if the carb is in good condition this will only be very occasionally.
I agree with you first move and off with it's head to see what you're dealing with first.
Changing the pistons is the best way to go, skimming the head helps get a better combustion chamber but does ultimately weaken it.
Raising compression will give more torque and punch, as well as better economy, the only potential downside is the risk of inducing detonation, dressing and polishing the combustion chambers would help prevent this as would higher octane fuel, an unrestrictive exhaust and a degree or two less ignition advance.
The SU will provide more power as it's nearer the size the engine needs, the Weber is too small. Topping up dashpots is no more hassle than checking the engine oil or radiator weekly.
I'd also recommend getting 3 angle valve seats cut and a large unrestrictive air filter like a K&N.