Partly you have answered your own query – every provider has its own systems, facilities and products that they offer. It is entirely down to them to decide what they wish to offer and put into the marketplace. If they do not wish to offer a TESSA ISA facility and are prepared to miss out on those potential funds, then that is their own decision.

If they do offer both facilities, they can keep the cash ISA and TESSA ISA funds separate – either via two distinct accounts or in one account while noting the proportions of each. If you subsequently wish to transfer either of the amounts, however, you may do so individually – provided that the provider allows you that flexibility. This is not necessarily always available.

If the new provider will not accept your previous TESSA ISA then you will have to find a separate home for it or leave it as is. It is also worth noting that, in practice, some receiving providers will not distinguish between a cash ISA and a TESSA ISA after receipt. So, the new receiving provider will only receive cash ISA funds (and be none the wiser).