I would have a word with Hargreaves Lansdown if I were you. I did have an ISA unit Trust I purchased from Fidelity but moved it to HL's vantage service some years back. Can't recall what happened though it all seemed straightforward.
With Cash ISAs you have to approach who you want it transferred to and they do the rest. I would have thought the same applied here.

I, too, would speak to H-L, they're very professional and helpful........they want your business and often do "free" transfer deals.

Indeed, I had to sell an ISA unit trust when transferring from Cofunds to Fidelity's funds supermarket. I do believe that also applies when one wants to leave Fidelity's funds supermarket. Yes, it seems unfairly restrictive, with the obvious aim at discouraging one from leaving either Cofunds or Fidelity. Saying that, you might also come out ahead of the game in a falling market.

The type of transfer where units are not sold and re-purchased is called "re-registraion" and is a "paper transaction" indeed. Many providers offer it and Fidelity FN are happy to move funds onto their platform this way but the last time I looked they were still reluctant to offer the same facility to their existing clients, i.e. to re-register away from Fidelity FN to another provider. You should enquire if they have started doing this, as there was talk that at some point they would have to relent.

If they won't do a re-registration transfer then that is a very good reason to avoid Fidelity FN.

As for Pat's transfer from HL Vantage to ISA: I did a similar thing with Barclays and they did not charge me commission. Barclays have a much lower commission for buying/selling shares than HL. HL are good for unit trusts but should be avoided for shares. I have ISAs and accounts with both HL and Barclays.