I'm sure others have said but just to summarise:
Your current ISA will still be an ISA but the rate will not be the same. As you worked out, you don't have to do anything. But as PPs observe, you can easily find an ISA with a better rate - close to 4%. I notice that the 1 year cash ISAs usually have the best rates.
Do NOT withdraw from your current ISA unless you need the money desperately.
Interest earned in an ISA is not taxable so do not include it on an annual tax form BUT if you withdraw from an ISA there can be tax to pay.
If you're not using the money, do keep it in an ISA account, either cash or stocks&shares, because any gains/interest will not be taxed (until you withdraw).
Google now (as per links suggested by PPs) and choose a new ISA to open after your current one expires.
After your current ISA matures, open a new one. Add any spare funds you can, up to £20k, and instruct your new ISA to roll the old ISA money into the new ISA. The new ISA bank will do all the work so that your old/current ISA stays in the ISA 'wrapper'.
Generally, stocks & shares ISAs give a better long term return but their value can fluctuate with the markets (FTSE/NYSE, etc).