The Nokia 3310 is rather an old phone - it's surprising that the battery even holds charge.
I see 3 possible problems here:
1) SIM voltage - you don't indicate how old your SIM is, but given that you're on GiffGaff, I'm assuming even the SIM in your "old" lost phone was GiffGaff-branded, so it's likely to be a reasonably modern SIM.
SIMs come in 5V, 3.3V and 1.8V formats.
I can't find any information on what voltage the 3310 provides to SIMs, but it may be that it's simply not producing the correct voltage to your SIM (most likely, an over-voltage).
You don't say whether or not you can make calls or even register on the network using the replacement SIM - this information would be helpful.
2) You say the message centre number is "fine" - I don't want to prejudge, but it may be that it looks fine to you and actually isn't (I'm often guilty of not being able to see the wood for the trees).
If you could post the message centre number exactly as it appears on the display, this would be helpful (I would expect it to be in the form +44xxxxxxxxxx)
3) Assuming you reported your phone lost to GiffGaff, it's likely that they put a bar on your subscription, to prevent you being charged for calls made by whoever finds the phone.
It's possible that the bar(s) haven't actually been lifted yet, despite you receiving and "activating" your SIM.
It's also possible that you might somehow have enabled SS (Supplementary Service) barring.
If your phone can register on the network, I would try finding the call bar settings in the 3310's menu (probably under "security" or "settings"), and try each of the query options, to see if any SS bars are enabled.
If you can't find the menu options, you could try the following USSD strings (note that both the handset menu options and these USSD strings both require the handset to have registered on the network using the GiffGaff SIM, since the bars are actually activated in the HLR and downloaded to the VLR):
*#33# BAOC (Bar All Outgoing Calls) Query
*#331# BOIC (Bar all Outgoing International Calls) Query
*#332# BOICXHC (Bar all Outgoing International Calls eXcept to Home Country) Query
*#35# BAIC (Bar All Incoming Calls) Query
*#351# BAICROAM (Bar All Incoming Calls when ROAMed) Query
If you haven't got SS barring enabled, then without being able to try your SIM in another handset (as opposed to other SIMs in the same handset), it's impossible from a user persepective to determine whether or not the fault is an ODB (Operator-Determined Bar) still present on your subscription, or a SIM voltage issue.
If you can post back here (using the "Additional Details" button), it would be helpful