Luck will not be on your side. The conversions will trigger gains but if plan these into 2 or more different UK tax years, the tax may not be too bad.
I'm really curious how the gains are calculated now and into the future.
Suppose for example I owned 1,000 shares of the index mutual fund VTSMX at $60/share, with a cost basis per share of $50/share giving unrealized gains of $10/share ($10,000). I now convert it to the ETF equivalent, VTI.
Vanguard do this with no US tax implications and now I own 500 shares at $120/share, with cost basis of $100/share, (these are the approximate current prices) with unrealized gains of $20/share ($10,000).
Will HMRC consider that I have actually sold the VTSMX shares and owe tax on a capital gain of $10,000? If so then I will need to keep my own records and know that my cost basis on date of conversion is $120/share. Ongoing the cost basis of my VTI shares is going to be different for the IRS and HMRC.
ETA
If this is the case then it would be better to actually sell the VTSMX shares and then buy the VTI shares. That way the HMRC tax can used as a credit against the US taxes and ongoing the accounting will be much easier.