I personally have no idea if the front shocks on a 1.2 Sports model are any different from a standard 1.2, but if they are, then you can buy individual shock absorbers from the famous auction site for as little as £37 each:
https://www.ebay.co.uk/itm/BOLK-Shock-Absorber-fits-Fiat-500-1-2-1-4-1-3-D-Multijet/311979008075?hash=item48a365e04b:m:mDqXDhlodZRfpLyID2KK3AQ
https://www.ebay.co.uk/itm/BOLK-Shock-absorber-sold-individually-FIAT-500-BOL-E061056/311918833212?hash=item489fcfae3c:g:QxIAAOSwXeJaupY4
or:
https://www.ebay.co.uk/itm/2X-GENUINE-MONROE-SHOCK-ABSORBER-GAS-FRONT-FIAT-500-C-07/382236348848?hash=item58ff0fd1b0:g:8-AAAOSwZapZyibB
Personally, I'm a fan of Bilstein shocks, you get what you pay for.
Living in London is not going to help you any on cost, one of the perks I guess of living in the Capital.
You could start by looking here:
https://www.goodgaragescheme.com/
Or you could search for a mobile mechanic whose overheads will be a little cheaper.
I'm guessing if you get a garage to do the job, you won't get much change out of £300. It all depends on what condition the top mounts and coil springs are in as well, which could see the price go up further. In any case, I would be reluctant to have just one side replaced, especially if the car has lots of miles on it. You should ideally always have suspension components replaced in pairs.
What year is your car, how many miles on it and who has told you that you require new shocks? Has the car failed its MoT?
I respect John202020's mechanical knowledge, which is probably, almost as good ast mine :devil:, but if you're not mechanically minded and you don't have the tools, then you have no choice but to get someone who knows what they are doing to do the job. And trust me, nothing ever goes exactly according to plan, they'll always be something that crops up that you weren't prepared for.